1 00:00:06,520 --> 00:00:09,110 OK Thank you Lou It's so. 2 00:00:10,939 --> 00:00:18,000 What makes what makes a great problem a really great mathematical problem well. 3 00:00:19,210 --> 00:00:22,319 Here are four factors I mean this is a good parlor game and 4 00:00:22,319 --> 00:00:23,960 you'll have your own answers. 5 00:00:25,350 --> 00:00:27,530 The first one is beauty I mean as. 6 00:00:28,680 --> 00:00:33,770 John Conway famously said it's a thing that non-mathematicians don't understand 7 00:00:33,770 --> 00:00:38,690 that mathematics is in a static field almost in its entirety so 8 00:00:38,690 --> 00:00:42,500 one of the things that's part of the development for 9 00:00:42,500 --> 00:00:45,870 you that are more junior actually all of you are more junior. 10 00:00:46,949 --> 00:00:47,449 And. 11 00:00:51,499 --> 00:00:52,880 Is that one of the things. 12 00:01:01,060 --> 00:01:05,850 But I mean one of the things as you're going through the junior ranks 13 00:01:05,850 --> 00:01:10,480 is you really you develop in this that it can having a static is really an essential 14 00:01:10,480 --> 00:01:15,660 part of mathematics The second thing is the history of the problem I mean problems 15 00:01:15,660 --> 00:01:21,099 are not great problems when they're first proposed they have to have a nice history 16 00:01:21,099 --> 00:01:25,989 of different kinds of events and it's even it's best if there are some partial 17 00:01:25,989 --> 00:01:31,020 results going along you know that the history is nobody did anything for 18 00:01:31,020 --> 00:01:35,279 three hundred years and then it was solved that that's not a good history the third 19 00:01:35,279 --> 00:01:41,680 thing is the connections to other areas that is if the proof uses. 20 00:01:43,599 --> 00:01:50,210 Interesting techniques and the fourth thing is afterwards solved 21 00:01:50,210 --> 00:01:55,010 does it have an effect are there follow ups to it so. 22 00:01:56,349 --> 00:02:01,010 THE ONE THAT'S MY FAVORITE problem and I was in high school 23 00:02:01,010 --> 00:02:06,099 I do wonderful teacher I grew up in Queens New York and I had a wonderful. 24 00:02:06,099 --> 00:02:08,140 I went to a regular public high school. 25 00:02:09,450 --> 00:02:14,230 And I but I had a wonderful high school teacher and one day he showed me the twin 26 00:02:14,230 --> 00:02:19,389 prime conjecture you know that there are infinite number of pairs of primes that 27 00:02:19,389 --> 00:02:25,500 are two apart and I had an epiphany in the pit Fanie isn't. 28 00:02:26,909 --> 00:02:32,090 Like like I realized that this is either true or 29 00:02:32,090 --> 00:02:37,319 false I mean it's just and it doesn't matter which universe 30 00:02:37,319 --> 00:02:42,179 we're live it doesn't matter if we're silicon based life forms or inverse Q. 31 00:02:42,179 --> 00:02:44,130 gravity you know it's either true or 32 00:02:44,130 --> 00:02:47,620 false it's as absolute a question as you can get. 33 00:02:48,720 --> 00:02:52,420 And we don't know the answer and we still don't know the answer thanks to 34 00:02:52,420 --> 00:02:57,150 you ten Jang and a lot of people before him we're closer 35 00:02:57,150 --> 00:03:01,120 to the answer now than we were before but we still don't know the answer to that so 36 00:03:01,120 --> 00:03:06,609 to me this is number one on my list it's a beautiful problem that 37 00:03:06,609 --> 00:03:11,220 that a high school student can understand the problem it's 38 00:03:11,220 --> 00:03:16,610 got a tremendous history with partial results dating back hundreds of years 39 00:03:16,610 --> 00:03:22,090 the methods have all kinds of interesting connections to analytic number theory. 40 00:03:23,190 --> 00:03:26,660 And the effect Well of course it hasn't been proven yet but 41 00:03:26,660 --> 00:03:32,349 already the effect on other areas is very strong so to me this one is number one. 42 00:03:34,970 --> 00:03:39,379 Now what about Merced San primes primes of the form two to the P. 43 00:03:39,379 --> 00:03:43,489 minus one are there an infinite number of Merson primes aired or 44 00:03:43,489 --> 00:03:44,930 she used to say you know. 45 00:03:46,199 --> 00:03:49,890 We know nothing about that and in one hundred years we're going to know nothing 46 00:03:49,890 --> 00:03:54,410 about that it's just too hard it's just too or 47 00:03:54,410 --> 00:04:00,919 no course you know maybe tomorrow you Tang will say I prove that to us OK So 48 00:04:00,919 --> 00:04:05,980 you know you can be wrong but I'll certainly give it a green light. 49 00:04:05,980 --> 00:04:06,900 For the beauty. 50 00:04:08,009 --> 00:04:11,090 And up to today with found forty eight of them 51 00:04:11,090 --> 00:04:13,530 you know we're working hard with the computer. 52 00:04:14,610 --> 00:04:19,440 But the problem the red lights are because there's really nothing 53 00:04:19,440 --> 00:04:24,420 it just seems to be a brick wall we're just not getting anywhere on proving or 54 00:04:24,420 --> 00:04:28,960 disproving that there are an infinite number of Merson primes I hope that's 55 00:04:28,960 --> 00:04:33,909 wrong but as of today that's how I see it the four color problem. 56 00:04:36,770 --> 00:04:40,880 Certainly was about as beautiful a problem as you can come up with that any plainer 57 00:04:40,880 --> 00:04:45,330 graph can be colored in four colors and it had a beautiful 58 00:04:45,330 --> 00:04:50,080 history dating back to the camp all that. 59 00:04:51,280 --> 00:04:56,730 Yeah all these partial results and really beautiful history and 60 00:04:56,730 --> 00:05:03,460 everything was really great until nine hundred seventy six because then Hakan and 61 00:05:03,460 --> 00:05:09,699 Apple proved that using hundreds and hundreds of cases and 62 00:05:09,699 --> 00:05:14,590 it was really an ugly proved and it's been improved but 63 00:05:14,590 --> 00:05:19,800 it's still ugly and surprisingly as a graph theorist it really 64 00:05:19,800 --> 00:05:25,140 hasn't had that much effect there is the fact of it is use but the methodology so 65 00:05:25,140 --> 00:05:29,290 I give it red lights for that part of. 66 00:05:30,540 --> 00:05:36,900 The room on hypothesis that there that there's zero zero of of 67 00:05:36,900 --> 00:05:42,129 the Riemann zeta function are on the critical line well 68 00:05:42,129 --> 00:05:46,830 tremendous history incredible connections because really 69 00:05:46,830 --> 00:05:51,879 it's saying that the primes are pseudo random I mean if you really understand it 70 00:05:51,879 --> 00:05:56,120 an incredible effect because the methods that have been used have really been 71 00:05:56,120 --> 00:05:59,919 I'll give it a yellow light mainly because I hate complex variables so. 72 00:06:00,989 --> 00:06:05,379 You know and so but you know you may have a different. 73 00:06:10,520 --> 00:06:14,210 Yes it is my personal statement OK OK. 74 00:06:15,339 --> 00:06:21,360 And for MA Well for ma is really number one for everybody and enormous history X. 75 00:06:21,360 --> 00:06:22,179 the N. plus Y. 76 00:06:22,179 --> 00:06:28,159 then equals Ian and when Wiles did prove it and 77 00:06:28,159 --> 00:06:33,450 people are still really working and all the connections to very deep connect 78 00:06:33,450 --> 00:06:38,339 very deep questions in algebraic geometry so firm wins the prize for 79 00:06:38,339 --> 00:06:42,060 everything but to me I also like twin primes but I think for 80 00:06:42,060 --> 00:06:46,400 most people from Oz number one but I'm going to propose another one. 81 00:06:48,190 --> 00:06:49,230 Our of three K.. 82 00:06:50,330 --> 00:06:51,290 Our of three K. 83 00:06:51,290 --> 00:06:52,530 that is what are the ass and 84 00:06:52,530 --> 00:06:57,019 topics of the Ramsey number three which I'll define in in just a moment. 85 00:06:58,569 --> 00:07:03,490 And I'll try to convince you why at least for 86 00:07:03,490 --> 00:07:06,830 me this gets the four green light so that's going to be. 87 00:07:08,040 --> 00:07:10,200 The topic of today. 88 00:07:13,921 --> 00:07:16,921 When George secular ash was. 89 00:07:22,799 --> 00:07:28,648 So much later in life he wrote something for 90 00:07:28,648 --> 00:07:32,879 about Paul erudition it was a very nice piece. 91 00:07:34,719 --> 00:07:38,598 And he described the paper that they wrote 92 00:07:39,598 --> 00:07:43,928 it was published in one nine hundred thirty five but it was really written. 93 00:07:45,129 --> 00:07:49,758 In one thousand nine hundred in the winter of one thousand thirty one thirty two and 94 00:07:49,758 --> 00:07:56,089 I really love this description of the life as a young man. 95 00:07:57,859 --> 00:07:58,559 In. 96 00:08:00,188 --> 00:08:02,609 In Budapest and. 97 00:08:04,618 --> 00:08:09,548 You know a lot of it is I think it it gives the joy of mathematics and 98 00:08:09,548 --> 00:08:12,469 it gives the joy of youth and 99 00:08:12,469 --> 00:08:18,039 I just like this quotation very much so I'll talk about the math in a moment. 100 00:08:19,539 --> 00:08:21,528 So let's define our of three K. 101 00:08:21,528 --> 00:08:22,418 not our three K. 102 00:08:22,418 --> 00:08:26,928 is usually defined as a min max but I really think of it there's an upper bound 103 00:08:26,928 --> 00:08:32,459 in a lower bound and then what you really want to do is sandwich you want 104 00:08:32,459 --> 00:08:36,829 them to come together so I really think of it as two problems upper bounds and 105 00:08:36,829 --> 00:08:42,239 lower bounds so to say that the Ramsey number is bigger than K. 106 00:08:42,239 --> 00:08:47,509 is talking about the existence of a graph and I tell the size existence because. 107 00:08:48,689 --> 00:08:53,709 I like errors magic So it's a natural thing and you want to graph and 108 00:08:53,709 --> 00:08:58,678 vertices no triangles and no independent set of size K. 109 00:08:58,678 --> 00:09:03,068 and if such a graph exists then you have the lower bound R. 110 00:09:03,068 --> 00:09:04,218 of three K. 111 00:09:04,218 --> 00:09:08,208 is greater than actually you can switch around like K. 112 00:09:08,208 --> 00:09:12,839 be a function of and and if you want and 113 00:09:12,839 --> 00:09:17,829 the upper bound would be to get an upper bound would say that if you take 114 00:09:17,829 --> 00:09:21,269 any graph any triangle all of this is from. 115 00:09:21,269 --> 00:09:24,448 Regress if you take any triangle for a graph with. 116 00:09:27,119 --> 00:09:34,179 And vertices then there you can find in again color red because 117 00:09:34,179 --> 00:09:39,659 how do you find it just means there exists an independent set of size K. 118 00:09:39,659 --> 00:09:42,588 so to my mind the asymptotic of our three K. 119 00:09:42,588 --> 00:09:47,168 are really two problems the upper bound and the lower bound. 120 00:09:48,809 --> 00:09:54,509 And here's the reference to air dish Klein and 121 00:09:54,509 --> 00:10:01,059 secular ash Klein had just come back from a summer school in Girton going and 122 00:10:01,059 --> 00:10:06,519 proposed a geometry problem and they all said about. 123 00:10:08,429 --> 00:10:13,339 Every dish was the baby of the group but he was already well known 124 00:10:13,339 --> 00:10:17,579 I think at this time well we can figure it out it was one thousand nine hundred 125 00:10:17,579 --> 00:10:21,868 thirty two he was born in one nine hundred thirteen so if you want to feel humble. 126 00:10:23,149 --> 00:10:25,239 He was one thousand years old at the time. 127 00:10:27,619 --> 00:10:31,919 And and they were a couple of years older and. 128 00:10:34,738 --> 00:10:38,077 And but if I were the results finally appeared. 129 00:10:39,558 --> 00:10:43,068 In a paper in one nine hundred thirty five and 130 00:10:43,068 --> 00:10:47,438 why there are only two authors may reflect the. 131 00:10:48,978 --> 00:10:54,438 Same gender stuff at the time I don't know leave that open was long long 132 00:10:54,438 --> 00:11:00,128 time ago and basically though there were a couple of proofs but 133 00:11:00,128 --> 00:11:05,778 in the proof they gave they rediscovered Ramses theorem and 134 00:11:05,778 --> 00:11:11,918 they gave in the proof it's implicit this upper bound for the Ramsey numbers. 135 00:11:13,518 --> 00:11:15,948 I don't know if the I don't think it was explicitly written but 136 00:11:15,948 --> 00:11:19,068 it was there in in the paper. 137 00:11:22,118 --> 00:11:23,927 And so in particular when L. 138 00:11:23,927 --> 00:11:28,588 equals three you get a big zero of case squared 139 00:11:28,588 --> 00:11:33,697 as the upper bound for our of three K. 140 00:11:33,697 --> 00:11:35,828 I mean I should say the general problem of our of L. 141 00:11:35,828 --> 00:11:37,178 K. is interesting too but 142 00:11:37,178 --> 00:11:39,508 today I'm only looking at our of three K.. 143 00:11:42,179 --> 00:11:47,718 It's called the happy ending problem because secular and Klein. 144 00:11:48,999 --> 00:11:54,519 Soon after got married and they had a very interesting history 145 00:11:54,519 --> 00:11:59,808 as the dark days descended on Europe they fled to. 146 00:12:01,079 --> 00:12:05,329 Shanghai Shanghai Shanghai had a very 147 00:12:05,329 --> 00:12:10,308 interesting history during World War two Very difficult to street 148 00:12:10,308 --> 00:12:15,069 your Chinese because it was taken over by the Japanese. 149 00:12:16,559 --> 00:12:21,709 But people fled from everywhere to Shanghai and it had 150 00:12:21,709 --> 00:12:26,879 already been an international community there was an international compound and 151 00:12:26,879 --> 00:12:31,499 so there were people from all over in Shanghai but it had a particular 152 00:12:31,499 --> 00:12:36,219 effect on the Jewish community and secular action Klein were Jewish. 153 00:12:37,438 --> 00:12:41,098 Because even when the Japanese 154 00:12:41,098 --> 00:12:45,389 took over Shanghai though the Japanese and the Germans were. 155 00:12:46,779 --> 00:12:52,478 Were allies the Japanese were not anti Jewish So the Jews not 156 00:12:52,478 --> 00:12:57,139 that things were easy I mean it was during the war it was difficult for everybody. 157 00:12:58,489 --> 00:13:04,978 But they they did live their life in Shanghai by the way i 158 00:13:04,978 --> 00:13:10,999 Movie review the movie last caution is one of my favorite movies and. 159 00:13:12,569 --> 00:13:17,289 It's Shanghai during that time actually it's in Chinese so it's. 160 00:13:18,959 --> 00:13:20,368 I don't know the Chinese title. 161 00:13:21,908 --> 00:13:27,478 And it takes place in Shanghai during the days of the Japanese but 162 00:13:27,478 --> 00:13:31,129 the only problem with that movie is there's no math in it but other but 163 00:13:31,129 --> 00:13:36,118 it is a good movie it is a good movie All right I want to look at 164 00:13:36,118 --> 00:13:41,949 things through an algorithmic lens and I think really. 165 00:13:41,949 --> 00:13:43,308 Yeah. 166 00:13:43,308 --> 00:13:44,548 About Lust Caution. 167 00:13:47,309 --> 00:13:48,609 But. 168 00:13:48,609 --> 00:13:52,939 And then they moved to Australia and indeed I have spoken to people especially. 169 00:13:54,679 --> 00:13:55,239 Two. 170 00:13:56,698 --> 00:13:59,989 To Nick warm all who said that they were really instrumental in 171 00:13:59,989 --> 00:14:04,739 setting up Australian mathematics in the post-war period so 172 00:14:04,739 --> 00:14:10,218 they really had wonderful long long lives and it was 173 00:14:10,218 --> 00:14:15,319 really wonderful so it really was a happy ending for them so 174 00:14:15,319 --> 00:14:19,209 I want to look at things algorithmically I mean I think that's the way. 175 00:14:20,598 --> 00:14:26,539 Computers algorithms I mean we look at pure mathematical problems but 176 00:14:26,539 --> 00:14:32,109 looking at them through the algorithmic lens I think is very useful So 177 00:14:32,109 --> 00:14:36,359 here's the an algorithm we take a graph on K. 178 00:14:36,359 --> 00:14:42,699 square points and I want to find an independent set of size K. 179 00:14:42,699 --> 00:14:46,448 so well if any point has degree K. 180 00:14:46,448 --> 00:14:50,569 I just look at its neighbors to see if it's triangle free the neighbors of 181 00:14:50,569 --> 00:14:56,309 a point are an independent set so if any point has degree K. 182 00:14:56,309 --> 00:15:01,209 I just take its neighbors I'm done else I pick a point throughout its neighbors 183 00:15:01,209 --> 00:15:05,439 pick a point throughout its neighbors pick a point throughout its neighbors 184 00:15:05,439 --> 00:15:08,079 each time I'm picking one point I'm throwing out K. 185 00:15:08,079 --> 00:15:13,969 minus one point so I get to pick a point and there are an independent set so 186 00:15:13,969 --> 00:15:18,408 this is the argument for case squared you can be a little bit more careful about it 187 00:15:18,408 --> 00:15:23,459 and you actually get the error to shoot down so 188 00:15:23,459 --> 00:15:28,959 it's not like it's a better proof than errors Secrest but I like algorithmic you. 189 00:15:29,999 --> 00:15:36,029 Now the lower bound came in a paper in the Bulletin of the American math society 190 00:15:36,029 --> 00:15:41,369 in one nine hundred forty seven A but I like to refer to one it was the. 191 00:15:41,369 --> 00:15:46,138 Gov and it's in the diaries of Paul Aaron ish and 192 00:15:46,138 --> 00:15:50,828 it was discovered in April one thousand nine hundred forty six which was a very 193 00:15:50,828 --> 00:15:56,668 important phase transition for me but no it was not when I got married OK. 194 00:15:57,919 --> 00:15:59,929 So leave you to figure that one out. 195 00:16:01,179 --> 00:16:04,859 And this is the argument that if enters K. 196 00:16:04,859 --> 00:16:09,638 two the one minus Kate choose two is less that one then indeed 197 00:16:10,909 --> 00:16:14,348 So this one is not for our of three K. 198 00:16:14,348 --> 00:16:15,499 it's for our of K. 199 00:16:15,499 --> 00:16:16,569 K. 200 00:16:16,569 --> 00:16:21,119 then there exists a graph without a clique or an independent set of size K. 201 00:16:21,119 --> 00:16:25,629 and the way we look at it now is we look at a random coloring and 202 00:16:25,629 --> 00:16:30,459 the probability of failure is less than one and 203 00:16:30,459 --> 00:16:35,249 so therefore the graph exists and this gives abound on our of cake 204 00:16:35,249 --> 00:16:39,619 Well I'd say more about that but we're really concentrating on our of three K. 205 00:16:39,619 --> 00:16:43,289 Well Eris wrote a lot of papers and. 206 00:16:44,438 --> 00:16:49,119 I think in terms of his legacy we can't look at all of them so 207 00:16:49,119 --> 00:16:53,349 I like to point out a couple of them that were particularly good and 208 00:16:53,349 --> 00:16:56,999 this one I think was really a remarkable paper 209 00:16:56,999 --> 00:17:01,768 mostly because of its date it was nine hundred sixty one and 210 00:17:01,768 --> 00:17:06,859 the probabilistic method was really just getting started and when you read this 211 00:17:06,859 --> 00:17:13,329 it's really a technical tour de force that he could do this at this early date and 212 00:17:13,329 --> 00:17:17,408 since then they've been many other arguments and he shows. 213 00:17:18,938 --> 00:17:25,379 A lower bound of case square over logs squared K. 214 00:17:25,379 --> 00:17:26,008 and. 215 00:17:27,069 --> 00:17:31,779 The argument he used counting arguments but in modern terms we 216 00:17:31,779 --> 00:17:36,579 look at random gress you look at a random graph with probability and 217 00:17:36,579 --> 00:17:41,369 it turns out the right probability to use is small constant on. 218 00:17:41,369 --> 00:17:46,859 Switching around the end in the case it's better to say and vertices no triangle and 219 00:17:46,859 --> 00:17:50,329 you want no independent of the size so instead of N. 220 00:17:50,329 --> 00:17:51,708 is a function of K. 221 00:17:51,708 --> 00:17:52,739 this is the K. 222 00:17:52,739 --> 00:17:56,019 as a function of and said OK and 223 00:17:56,019 --> 00:18:01,749 the idea was that he looked at a random graph the right thing to look at was P. 224 00:18:01,749 --> 00:18:06,919 is epsilon and to the minus one half where epsilon was a small constant now that 225 00:18:06,919 --> 00:18:12,449 graph will have lots of triangles in fact it will have epsilon cubed and 226 00:18:12,449 --> 00:18:17,359 to the three have strangles But then what he says 227 00:18:17,359 --> 00:18:22,428 is that basically 228 00:18:22,428 --> 00:18:28,308 you can eliminate the triangles you can eliminate the triangles so what he says. 229 00:18:34,359 --> 00:18:38,808 Is that for 230 00:18:38,808 --> 00:18:43,738 every case set off by not that there isn't a triangle 231 00:18:45,787 --> 00:18:49,138 but every time there is a triangle 232 00:18:52,647 --> 00:19:00,247 that contains no sart he says that these case sets. 233 00:19:02,488 --> 00:19:07,418 We want no independence of size K. 234 00:19:07,418 --> 00:19:11,988 and we don't get it immediately but we say that that. 235 00:19:13,908 --> 00:19:17,268 Every independence that has a. 236 00:19:18,918 --> 00:19:21,918 An edge that cannot 237 00:19:24,689 --> 00:19:28,829 be extended to a triangle technically a triangle outside. 238 00:19:29,958 --> 00:19:34,019 And the thing is and then he proves that it's not at all easy 239 00:19:34,019 --> 00:19:38,749 that the random graph will will have this property with high probability and 240 00:19:38,749 --> 00:19:43,278 his real tour de force just to do that but then comes the. 241 00:19:44,289 --> 00:19:48,579 So the air dish magic is you get a graph that for every case that. 242 00:19:51,918 --> 00:19:57,249 There is an edge there that might be in triangles but 243 00:19:57,249 --> 00:20:03,119 it's not extended and you can't extend it to a triangle outside 244 00:20:03,119 --> 00:20:06,208 once you have that graph just take that graph and 245 00:20:06,208 --> 00:20:12,329 apply a greedy algorithm not a random just put the edges in any order and 246 00:20:12,329 --> 00:20:17,699 just accept them if they don't form a triangle when you get to this add. 247 00:20:19,669 --> 00:20:24,538 You see if it does form a triangle it would have to have been inside and 248 00:20:24,538 --> 00:20:29,289 so you would have edges there so this is the argument and 249 00:20:29,289 --> 00:20:33,589 then it involves this greedy algorithm but 250 00:20:33,589 --> 00:20:38,549 again the fact that he did this in one thousand nine hundred sixty one was really 251 00:20:38,549 --> 00:20:43,939 remarkable and since then have been many other arguments OK Let 252 00:20:43,939 --> 00:20:49,668 me now turn to the lower bound so you recall. 253 00:20:51,908 --> 00:20:57,079 I mean upper bound so this should be upper all depends on whether NS A function of K. 254 00:20:57,079 --> 00:21:02,679 or case a function so the original upper bound wished to go of case squared and 255 00:21:02,679 --> 00:21:07,659 for a long time it was open whether it was a little lower case squared and 256 00:21:07,659 --> 00:21:09,789 this was solved in one thousand nine hundred sixty eight. 257 00:21:10,879 --> 00:21:16,559 By graver and Jaco but mathematics is such a tough field sometimes 258 00:21:16,559 --> 00:21:21,449 you can really work and get a great result and then later somebody gets a go so 259 00:21:21,449 --> 00:21:25,239 nobody knows about braver and Jaco I mean I just threw it in but I'm not even going 260 00:21:25,239 --> 00:21:30,749 to show the proof it's a method to use bad ones yeah bad ones Ya know I once I 261 00:21:30,749 --> 00:21:37,059 knew the problem ya know is a complicated really messy Pruitt probably because you 262 00:21:37,059 --> 00:21:42,069 know it well they didn't use probabilistic language maybe I could go back but 263 00:21:42,069 --> 00:21:45,349 I'm not going to OK but then. 264 00:21:46,518 --> 00:21:52,429 The good proof was by lotion some ready in nineteen. 265 00:21:53,459 --> 00:21:56,408 Eighty that remove the log log K. 266 00:21:56,408 --> 00:22:00,639 factor and I remember at the time I take home 267 00:22:00,639 --> 00:22:04,779 at least somebody who's got a funny sense of humor anyway so it's hard to tell but 268 00:22:04,779 --> 00:22:08,959 he was very disappointed he said it only approved by a log log Cait term but 269 00:22:08,959 --> 00:22:15,209 as we'll see it turned out to be a big thing and so they get this argument. 270 00:22:16,239 --> 00:22:19,229 That it's case squared over log OK. 271 00:22:19,229 --> 00:22:23,478 And what it really is is that if you have an vertices and 272 00:22:23,478 --> 00:22:28,049 no triangle and if the average degree is K. 273 00:22:28,049 --> 00:22:32,059 then you get an independent set which is not just an overcast day 274 00:22:32,059 --> 00:22:34,488 after all of the maximum degrees K. 275 00:22:34,488 --> 00:22:36,819 you'll get an independent said event over K. 276 00:22:36,819 --> 00:22:41,619 tautologically But there's this extra factor of law that because 277 00:22:41,619 --> 00:22:46,828 there's no triangle you not only get the independent set of size and over K. 278 00:22:46,828 --> 00:22:48,648 you get a bigger independents. 279 00:22:49,769 --> 00:22:51,148 And the idea. 280 00:22:52,349 --> 00:22:52,889 Again. 281 00:22:54,069 --> 00:22:58,839 Doing a lot of handwaving is that you're going to erred a typical 282 00:22:58,839 --> 00:23:03,848 vertex that you're going to add vertices to the independent set one by one. 283 00:23:05,979 --> 00:23:06,479 And. 284 00:23:07,919 --> 00:23:12,868 So let me introduce the differential equation and all of you that know Lutes I 285 00:23:12,868 --> 00:23:18,199 hope you all know this you know so so here's the idea 286 00:23:18,199 --> 00:23:22,809 of the differential equation suppose that you've picked an over K. 287 00:23:22,809 --> 00:23:23,929 times T.. 288 00:23:24,989 --> 00:23:29,929 Points already and one of the things I've learned for the last. 289 00:23:31,149 --> 00:23:35,018 Quite a few years I've been going to Microsoft to the group led by 290 00:23:35,018 --> 00:23:39,549 Jennifer Chazen Christian boards and are all these mathematical physicists and 291 00:23:39,549 --> 00:23:43,649 of I used to be very disdainful of them but they're not so 292 00:23:43,649 --> 00:23:48,198 bad after all and I've learned some things from them. 293 00:23:49,639 --> 00:23:53,419 And one of the things I learned was they have this view that the key thing to 294 00:23:53,419 --> 00:23:54,209 do is. 295 00:23:55,489 --> 00:24:02,448 That when you have a problem how do you scale it in the right way and 296 00:24:02,448 --> 00:24:06,628 it's hard I mean usually the answer is first you do the wrong scaling and 297 00:24:06,628 --> 00:24:08,829 then you find out it's not working and 298 00:24:08,829 --> 00:24:12,429 then you find the right scale in this case the right scale and 299 00:24:12,429 --> 00:24:16,259 it makes it hard to read papers because when you read the papers the papers 300 00:24:16,259 --> 00:24:19,229 always put in I mean why should they put in the wrong scale of course they put. 301 00:24:19,229 --> 00:24:21,188 They put in the right scale but 302 00:24:21,188 --> 00:24:25,379 when you're trying to do something you don't know the scaling at first and so it 303 00:24:25,379 --> 00:24:30,999 makes doing it much more difficult turns out the right scaling is and over a K. 304 00:24:30,999 --> 00:24:31,999 times T.. 305 00:24:33,449 --> 00:24:36,838 Points are chosen and 306 00:24:36,838 --> 00:24:42,460 at that point you've chosen an over K. 307 00:24:42,460 --> 00:24:44,240 times T.. 308 00:24:46,080 --> 00:24:46,789 Points. 309 00:24:51,190 --> 00:24:54,060 Except until it isn't until it is it yeah and 310 00:24:54,060 --> 00:24:59,010 that's part of the him waving Well sorry what was. 311 00:25:03,630 --> 00:25:07,009 S. is the number of points remaining So 312 00:25:07,009 --> 00:25:10,120 as you pick a point you're throwing out the so 313 00:25:10,120 --> 00:25:13,860 you pick a point you throw out the neighbors OK So so 314 00:25:13,860 --> 00:25:19,720 you're at is at a certain stage the number of points remaining is S. 315 00:25:19,720 --> 00:25:22,720 a T. times and but critically are not going to 316 00:25:22,720 --> 00:25:26,459 stop at Epsilon and I have to keep going and that makes the proof harder. 317 00:25:27,880 --> 00:25:30,039 But so like one there and 318 00:25:30,039 --> 00:25:34,999 over two points remaining originally the maximum degree was K. 319 00:25:34,999 --> 00:25:38,880 if there are any over two points you might think that the average degree is K. 320 00:25:38,880 --> 00:25:45,960 over two and in fact this can be done so what happens is that let's think of S. 321 00:25:45,960 --> 00:25:46,680 of S. of T. 322 00:25:46,680 --> 00:25:51,110 as a half when half the points are remaining on that graph 323 00:25:51,110 --> 00:25:53,409 the average degree instead of being. 324 00:25:54,639 --> 00:25:55,959 K. 325 00:25:55,959 --> 00:25:56,520 is K. 326 00:25:56,520 --> 00:26:00,959 over two because you only have half the points and that's not a theorem and 327 00:26:00,959 --> 00:26:05,319 indeed you have to prove that you can find such a point but 328 00:26:05,319 --> 00:26:08,609 let's just assume the average degree is K. 329 00:26:08,609 --> 00:26:10,740 times so of T. 330 00:26:10,740 --> 00:26:14,530 then you pick a point with degree K. 331 00:26:14,530 --> 00:26:15,149 times S. 332 00:26:15,149 --> 00:26:16,060 over T. 333 00:26:16,060 --> 00:26:21,780 and you throw away all its neighbors so instead of throwing away K. 334 00:26:21,780 --> 00:26:23,880 point you're only deleting K. 335 00:26:23,880 --> 00:26:27,350 times as fifty points so what happens is. 336 00:26:29,300 --> 00:26:33,070 What is the number of points deleted it's K. 337 00:26:33,070 --> 00:26:33,719 times. 338 00:26:35,840 --> 00:26:36,859 K. times S. 339 00:26:36,859 --> 00:26:40,960 of T. but then when we do the normalization 340 00:26:40,960 --> 00:26:44,950 because time is being normalized by this so we're increasing time by K. 341 00:26:44,950 --> 00:26:49,069 overrun and what we get is this differential equation and 342 00:26:49,069 --> 00:26:54,899 then we wave our hands and we get this differential equation that S prime of T. 343 00:26:54,899 --> 00:26:56,899 is minus S. 344 00:26:56,899 --> 00:26:58,380 of T. 345 00:26:58,380 --> 00:27:01,410 and I'm not very good at differential equations but 346 00:27:01,410 --> 00:27:03,279 I know the answer to this one. 347 00:27:03,279 --> 00:27:05,299 It's like eat of the minus T. 348 00:27:05,299 --> 00:27:08,699 but then to answer Dana's question we don't just take T. 349 00:27:08,699 --> 00:27:13,280 constant So now we wave our hands and say well this is going to keep on going 350 00:27:13,280 --> 00:27:17,750 keep on going I mean they didn't wave their hands they they did it right and 351 00:27:17,750 --> 00:27:18,660 you say well it's S. 352 00:27:18,660 --> 00:27:19,760 to the minus T. 353 00:27:19,760 --> 00:27:22,890 so it should keep on going until T. 354 00:27:22,890 --> 00:27:23,390 is. 355 00:27:25,160 --> 00:27:29,920 Log because when tears log the. 356 00:27:31,850 --> 00:27:34,760 The process stops you end up with 357 00:27:34,760 --> 00:27:39,740 why does it stop because then you're down to an over caver to seize and. 358 00:27:41,960 --> 00:27:44,210 And then all the points just have degree one or 359 00:27:44,210 --> 00:27:48,290 so and to me you could even throw them in there only and over K. 360 00:27:48,290 --> 00:27:49,560 points so it doesn't. 361 00:27:52,718 --> 00:27:53,959 Yes of T. 362 00:27:53,959 --> 00:27:54,499 S. of T. 363 00:27:54,499 --> 00:27:56,128 becomes one over K. 364 00:27:56,128 --> 00:28:00,999 and that's when the process stops so what happens is I mean what 365 00:28:00,999 --> 00:28:05,959 I think of it as you go down to half the point the way I think of this is you go 366 00:28:05,959 --> 00:28:11,339 down to half til half the points are left it's like you're taking an over to T. 367 00:28:11,339 --> 00:28:12,359 so N. over to K. 368 00:28:12,359 --> 00:28:16,059 point but now you've got half the points and 369 00:28:16,059 --> 00:28:18,879 over two points but now the average agrees K. 370 00:28:18,879 --> 00:28:23,809 over two so now when you again take half the points you go down to end 371 00:28:23,809 --> 00:28:28,089 over two over kayo Vertu so you getting another and over a K. 372 00:28:28,089 --> 00:28:32,719 points it's another way of looking at this differential good and you do this log K. 373 00:28:32,719 --> 00:28:36,399 times and so this is the intuition behind it. 374 00:28:37,769 --> 00:28:38,269 Here. 375 00:28:39,699 --> 00:28:44,369 Yes And sure looked at this well I don't know shearers very smart and 376 00:28:44,369 --> 00:28:49,399 I'm not sure how he did it but if you're very very careful with this you can get it 377 00:28:49,399 --> 00:28:54,469 more precise I'm being a little with this problem here is that. 378 00:28:56,968 --> 00:29:01,429 Yeah and I think even with the right constant even with this diversity. 379 00:29:03,909 --> 00:29:04,438 You know. 380 00:29:05,979 --> 00:29:10,849 That I'm not sure Yeah Yeah Yeahs to be. 381 00:29:21,309 --> 00:29:24,839 OK Yeah Jim sure looked at this and 382 00:29:24,839 --> 00:29:28,559 got very careful things with the right constant. 383 00:29:30,319 --> 00:29:30,929 But let's. 384 00:29:32,978 --> 00:29:33,478 OK. 385 00:29:35,569 --> 00:29:36,948 Let me turn to. 386 00:29:38,139 --> 00:29:43,128 This I mentioned the colloquium the globe us local lemma. 387 00:29:45,028 --> 00:29:50,549 If you have a vents AI I said it in a different way 388 00:29:50,549 --> 00:29:55,099 at the colloquium and they have a certain probability but 389 00:29:55,099 --> 00:30:01,249 you know that each AI is independent of all but a small number of other events and 390 00:30:01,249 --> 00:30:04,889 here I mean independent in the sense of mutually independent that is you take 391 00:30:04,889 --> 00:30:09,159 all these other events and it's mutually independent of that set except for 392 00:30:09,159 --> 00:30:15,069 the small number of events that it is dependent on and then the local lemma says 393 00:30:15,069 --> 00:30:20,299 that under certain conditions you can think of the local them as a sieve that 394 00:30:20,299 --> 00:30:26,038 there is a point in the probability space where none of these bad events occur and 395 00:30:26,038 --> 00:30:29,269 sometimes you get a result cheaply and 396 00:30:29,269 --> 00:30:34,028 that was my case here because I heard about. 397 00:30:35,229 --> 00:30:41,929 The low bus local Emma and so I thought well let's look at it for our three K. 398 00:30:41,929 --> 00:30:47,039 and so we have we put it in with probability P. 399 00:30:47,039 --> 00:30:51,318 and the bad events are of two types that you have a triangle and 400 00:30:51,318 --> 00:30:56,998 that you have an independent set of size OK So we have the variables 401 00:30:56,998 --> 00:31:03,869 we want to optimize we want to optimize and as a function of K. 402 00:31:03,869 --> 00:31:05,929 such that there exists a P. 403 00:31:05,929 --> 00:31:11,689 So if the local lemma will apply and so it was some very interesting 404 00:31:11,689 --> 00:31:16,729 calculus really quite involved was provide proud of myself to be able 405 00:31:16,729 --> 00:31:21,399 to get it but in the end I was a little disappointed because I got the air dish 406 00:31:21,399 --> 00:31:26,308 result of one thousand nine hundred sixty one but it gave another proof for the. 407 00:31:27,348 --> 00:31:29,109 For the air dish result. 408 00:31:37,069 --> 00:31:40,008 Yes that's right you get other results for our four K. 409 00:31:40,008 --> 00:31:40,508 that weren't. 410 00:31:42,199 --> 00:31:47,609 Getting that long yeah yeah so there are other cases in this case it 411 00:31:47,609 --> 00:31:52,749 duplicated the result and for our K.K. I have to men this 412 00:31:52,749 --> 00:31:57,708 is perhaps my most quoted paper for our K.K. If you just plug in the last local M. 413 00:31:57,708 --> 00:32:02,638 you improve the previous bound by a factor of two and I wrote that I mean it 414 00:32:02,638 --> 00:32:07,328 was just the simplest calculation I think that's my most referenced paper so some. 415 00:32:11,919 --> 00:32:13,239 The bounds are K. 416 00:32:13,239 --> 00:32:13,989 cubed in K. 417 00:32:13,989 --> 00:32:16,298 to the five halves if I. 418 00:32:17,359 --> 00:32:22,999 Say Yeah that's right you know there are four K. 419 00:32:22,999 --> 00:32:26,098 will go I think it will take another generation of we 420 00:32:26,098 --> 00:32:29,379 need some smart young people to look at our four K.B. 421 00:32:29,379 --> 00:32:32,679 blocking man this is like the logarithmic improvement but. 422 00:32:34,109 --> 00:32:37,048 Yeah right well that's true our four K. 423 00:32:37,048 --> 00:32:40,139 is way way open it's between K. 424 00:32:40,139 --> 00:32:42,249 to the five halves and K.Q. and 425 00:32:42,249 --> 00:32:47,768 it it's really a mystery that one is really a mystery. 426 00:32:50,489 --> 00:32:54,439 So then we turn to one thousand nine hundred ninety five and 427 00:32:54,439 --> 00:32:57,049 I think of this is a very romantic story. 428 00:32:58,149 --> 00:33:01,279 I mean here let's see her dish was born in one thousand and 429 00:33:01,279 --> 00:33:06,339 thirteen so a little calculation reveals that. 430 00:33:07,929 --> 00:33:09,869 Thirteen ninety five minus. 431 00:33:10,978 --> 00:33:12,759 He wasn't really a young man. 432 00:33:14,619 --> 00:33:16,199 At that time and 433 00:33:16,199 --> 00:33:21,439 he returned to the problem that he first worked on when he was nineteen years old. 434 00:33:22,478 --> 00:33:28,188 He came back to it and this was work with Peter Winkler and 435 00:33:28,188 --> 00:33:31,709 so when who have lost track of it he was in Florida somewhere. 436 00:33:33,339 --> 00:33:37,508 It appeared in one thousand I think it was actually the work was a couple years later 437 00:33:37,508 --> 00:33:41,308 but he said How about random read so 438 00:33:41,308 --> 00:33:45,229 what is the random greedy algorithm you're given. 439 00:33:46,488 --> 00:33:50,589 What you do is you start out with the and this is going to be a main 440 00:33:50,589 --> 00:33:54,448 part of the rest of the talk you start out with the empty graph and 441 00:33:54,448 --> 00:34:00,629 you put in edges at random except that if they form a triangle you reject them and 442 00:34:00,629 --> 00:34:04,859 that's it that's all you do so it's a random process and 443 00:34:04,859 --> 00:34:10,459 we call that the random greedy a random greedy triangle free process and 444 00:34:10,459 --> 00:34:14,399 the thing is you want to analyze this process well. 445 00:34:15,978 --> 00:34:21,499 Algorithms are hard to analyze because what happens today 446 00:34:21,499 --> 00:34:23,379 depends on what happened yesterday. 447 00:34:24,389 --> 00:34:28,959 But they did do some analysis and it and 448 00:34:28,959 --> 00:34:33,539 they have a nice paper they improve the constant. 449 00:34:34,849 --> 00:34:41,228 It would have been a wonderful story if if they actually solved our of three K. 450 00:34:41,228 --> 00:34:45,679 you know some sixty years after first working but it's still a wonderful story I 451 00:34:45,679 --> 00:34:50,119 mean it's still a wonderful story because what they did was they broke the ice for. 452 00:34:50,119 --> 00:34:53,599 Or the the final attack on it and 453 00:34:53,599 --> 00:34:58,138 I was able to look at their results and improve our of three K. 454 00:34:58,138 --> 00:35:01,909 by an arbitrary constant and 455 00:35:01,909 --> 00:35:07,058 I never actually worked on just making the constant a function of K. 456 00:35:07,058 --> 00:35:12,209 because and it never got appeared until it actually appeared in 457 00:35:12,209 --> 00:35:15,048 birthday thing I put it in there so. 458 00:35:16,819 --> 00:35:20,538 I like it but you know but there's a reason why. 459 00:35:21,728 --> 00:35:23,569 I mean why it's. 460 00:35:25,359 --> 00:35:33,758 Because I almost the same time John Hunt Kim was a post-doc at. 461 00:35:35,359 --> 00:35:40,309 Bell Laboratories and he had gotten his Ph D. 462 00:35:40,309 --> 00:35:45,319 from Jeff Kahn and he was one of the first people to really work on 463 00:35:45,319 --> 00:35:50,549 Martin gales in the common a Tauriel setting and 464 00:35:50,549 --> 00:35:55,139 he looked at this and he solved the problem there's up the constant He 465 00:35:55,139 --> 00:36:00,469 showed that this lower bound of case squared over log squared K. 466 00:36:00,469 --> 00:36:02,849 could be improved to K. 467 00:36:02,849 --> 00:36:04,588 squared over log K. 468 00:36:04,588 --> 00:36:09,609 which matches the upper bound due to a cold motion and 469 00:36:09,609 --> 00:36:15,749 Sam ready so this was his paper the Ramsey number our freak has or I don't 470 00:36:15,749 --> 00:36:20,139 know why you use teams that have Kerry but that's OK Has order of magnitude T. 471 00:36:20,139 --> 00:36:22,329 squared over log. 472 00:36:23,408 --> 00:36:28,438 It used it wasn't quite random greedy it used kind of a nibble method or 473 00:36:28,438 --> 00:36:31,588 it was sometimes called the semi random method. 474 00:36:33,188 --> 00:36:33,929 And. 475 00:36:35,209 --> 00:36:39,949 Used Martin gales in a very heavy way and different it used a whole lot of 476 00:36:39,949 --> 00:36:44,739 things he was awarded the focus and Prize for this in one nine hundred 477 00:36:44,739 --> 00:36:49,728 ninety seven it was really it was really the end of the story. 478 00:36:51,069 --> 00:36:51,938 Except it wasn't. 479 00:36:53,018 --> 00:36:56,768 Maybe it was the climax of the story but it wasn't the end of the story 480 00:36:56,768 --> 00:37:00,178 we all thought it was the end of the story because things kept on going. 481 00:37:02,190 --> 00:37:05,550 But I want to mention a personal moment so 482 00:37:05,550 --> 00:37:10,480 I think George seculars was born I believe it was nine hundred twelve. 483 00:37:11,940 --> 00:37:16,400 Maybe eleven and Esther and George were about the same age. 484 00:37:17,519 --> 00:37:20,840 So you can see they were not exactly young and 485 00:37:20,840 --> 00:37:26,150 I was in Australia and I've got to go back that was less than it was it is 486 00:37:26,150 --> 00:37:30,189 twenty years ago I'm going on and I gave this lecture. 487 00:37:31,220 --> 00:37:34,820 Except I mean you can see that the I keep changing the number I 488 00:37:34,820 --> 00:37:38,469 mean it's the same lecture I had slides and change the number. 489 00:37:40,579 --> 00:37:43,969 But about our three K. 490 00:37:43,969 --> 00:37:47,780 and it was just such a memorable thing because they're in the front row and 491 00:37:47,780 --> 00:37:54,570 still very lively an active and wonderful spirit were asked their client and 492 00:37:54,570 --> 00:37:59,230 George Secor If so it's a wonderful personal moment for 493 00:37:59,230 --> 00:38:04,660 me but the story didn't end so. 494 00:38:07,610 --> 00:38:08,611 Tom Bowman. 495 00:38:10,550 --> 00:38:18,289 Then showed that actually random greedy not semi random but just random greedy. 496 00:38:19,350 --> 00:38:23,670 Does work that is you just put in the edges at random and 497 00:38:23,670 --> 00:38:27,480 then what's going to happen is you're going to get. 498 00:38:28,950 --> 00:38:30,320 No independent. 499 00:38:31,450 --> 00:38:37,389 Of size large constant times Ruden log in 500 00:38:37,389 --> 00:38:42,650 now actually to me this random triangle random crying Go 501 00:38:42,650 --> 00:38:47,760 random try trying a free random greedy or however you want to permute the words is 502 00:38:47,760 --> 00:38:53,060 really the the motivation is is from Ramsey numbers but 503 00:38:53,060 --> 00:38:58,470 I find it a fascinating process to look at itself I mean 504 00:38:58,470 --> 00:39:05,290 how can we understand what this process what's happening in this process and 505 00:39:05,290 --> 00:39:11,389 it turns out that the key to looking at it is the differential 506 00:39:11,389 --> 00:39:16,259 equation method so you get some variables. 507 00:39:17,280 --> 00:39:21,899 At any stage you have to have the right parameterization and 508 00:39:21,899 --> 00:39:24,399 it turns out that you look at when T. 509 00:39:24,399 --> 00:39:25,609 times and over and 510 00:39:25,609 --> 00:39:31,109 to the three have spares have been accepted and then it's a function of T. 511 00:39:31,109 --> 00:39:35,120 but just like before in the N.T. is going to be a function of and 512 00:39:35,120 --> 00:39:38,300 the slow growing from a lot poly log function to Ben and 513 00:39:38,300 --> 00:39:41,020 that makes the differential equation method much harder. 514 00:39:43,500 --> 00:39:45,850 And then you have some statistics. 515 00:39:47,289 --> 00:39:50,640 So at any point pairs are in three there's. 516 00:39:51,800 --> 00:39:55,990 A trichotomy on the pairs either a pair is already in or 517 00:39:55,990 --> 00:40:00,030 you can't put it in because it would make a triangle or 518 00:40:00,030 --> 00:40:04,960 you can put it in but it hasn't come up yet OK so 519 00:40:04,960 --> 00:40:08,810 we call those in open and closed and 520 00:40:08,810 --> 00:40:14,499 then we have of riot statistics given to Verna Cs X. 521 00:40:14,499 --> 00:40:15,170 and Y.. 522 00:40:21,779 --> 00:40:29,030 X A Y U N V For example 523 00:40:34,489 --> 00:40:39,300 we have a statistic we can count the number of W. 524 00:40:39,300 --> 00:40:43,770 where they're both open we can count the number of. 525 00:40:45,370 --> 00:40:46,400 W.. 526 00:40:47,780 --> 00:40:48,519 Where. 527 00:40:55,551 --> 00:41:00,921 Where one of them is open and one of them is already in the graph OK And 528 00:41:00,921 --> 00:41:07,001 then we there's a global statistic about the number of vertices and then we have. 529 00:41:08,011 --> 00:41:10,961 Some scaling this turns out to be the right scale. 530 00:41:12,751 --> 00:41:15,411 And notice just to make things more difficult it is this is 531 00:41:15,411 --> 00:41:18,431 really enter used to statistics because for 532 00:41:18,431 --> 00:41:23,501 every human being you have this number and then you risk thickly. 533 00:41:24,551 --> 00:41:27,551 You wind up with these differential equations so 534 00:41:27,551 --> 00:41:31,711 I won't go into just you know how you get them but. 535 00:41:33,271 --> 00:41:36,960 Two You're looking at expected value and 536 00:41:36,960 --> 00:41:41,640 when you had so you adding one more edge so you're increasing T. 537 00:41:41,640 --> 00:41:44,871 by the appropriately parameterize time and 538 00:41:44,871 --> 00:41:49,600 now you want to look at the expected change in X. 539 00:41:49,600 --> 00:41:50,521 of T. 540 00:41:50,521 --> 00:41:52,611 and you wind up getting these. 541 00:41:53,651 --> 00:41:58,241 Differential equations and these differential equations have a nice 542 00:41:58,241 --> 00:42:03,850 solution and what happens is in a way that maybe still there should be a simpler 543 00:42:03,850 --> 00:42:08,831 solution reason for this the solution to the differential equations 544 00:42:08,831 --> 00:42:14,920 is just as if you would put in a random graph with that many edges and 545 00:42:14,920 --> 00:42:20,000 yet this is not a random graphic as it has no triangle and yet these statistics 546 00:42:20,000 --> 00:42:25,460 are the same and I think there's a deeper reason for it that I don't understand but 547 00:42:25,460 --> 00:42:31,041 you get the differential equations and and you solve them OK. 548 00:42:33,059 --> 00:42:37,890 And again I'm going to skip there's some stuff about why it should be but 549 00:42:37,890 --> 00:42:45,370 you get these things but 550 00:42:45,370 --> 00:42:50,370 now what you need is the differential equation method so 551 00:42:50,370 --> 00:42:56,680 you have these things in terms of expectation and now you'd like to say that 552 00:42:56,680 --> 00:43:02,979 with high probability the when you do the process you stick 553 00:43:02,979 --> 00:43:08,870 to the solution to the differential equation and this is something 554 00:43:08,870 --> 00:43:13,580 to me there are two the two figures on this our Nic were mauled and now. 555 00:43:15,300 --> 00:43:19,449 And I think this is a very powerful methodology this differential equation 556 00:43:19,449 --> 00:43:25,589 method that to understand a random process for asymptotically 557 00:43:25,589 --> 00:43:31,350 as the number of vertices goes to infinity you look at expectations in the so 558 00:43:31,350 --> 00:43:36,479 the process is just in tiny steps and you look at expectation 559 00:43:36,479 --> 00:43:41,339 you create a differential equation and then you want to show. 560 00:43:43,400 --> 00:43:48,190 That you're following the differential equation and the big problem as I 561 00:43:48,190 --> 00:43:53,020 see it here is the stability of the differential equation 562 00:43:53,020 --> 00:43:58,000 because as you're going along of course you're going to have error you know you're 563 00:43:58,000 --> 00:44:02,260 not going to follow the differential equation Exactly it's a random process so 564 00:44:02,260 --> 00:44:05,220 you're going to be a little bit off but the problem is if you're a little bit off 565 00:44:05,220 --> 00:44:11,760 you know maybe it gets worse and worse and so this is a natural question 566 00:44:11,760 --> 00:44:16,890 of what is the stability of the differential equation and. 567 00:44:18,010 --> 00:44:20,180 And so that's the first problem and 568 00:44:20,180 --> 00:44:24,280 then the second problem is it's not enough to do it for T. 569 00:44:24,280 --> 00:44:26,290 constant you want T. 570 00:44:26,290 --> 00:44:31,270 to go all the way up in order to get Kim's result you wanted to go up 571 00:44:31,270 --> 00:44:36,249 to constant squared of logon and I mean that turns out to 572 00:44:36,249 --> 00:44:41,130 be the the right parameterization And so that makes it much more difficult because 573 00:44:41,130 --> 00:44:45,470 there are a lot of theorems that say that you know if you have something it 574 00:44:45,470 --> 00:44:49,330 will follow the differential equation but most of the theorems 575 00:44:49,330 --> 00:44:52,379 you know you're just going from teakwood zero to Teague was a million. 576 00:44:53,440 --> 00:44:58,269 But we're not going FROM we're going on beyond infinity here we're going from 577 00:44:58,269 --> 00:45:02,900 teakwood zero to teak was epsilon squared of logon So 578 00:45:02,900 --> 00:45:06,470 just saying well I looked up a book on differential It was so 579 00:45:06,470 --> 00:45:11,850 therefore it works no you need stronger methodology and this is what 580 00:45:11,850 --> 00:45:17,160 we're Malden and loots have developed and I think it's very very powerful. 581 00:45:21,259 --> 00:45:23,079 You have Martin gales. 582 00:45:24,410 --> 00:45:24,910 But. 583 00:45:28,749 --> 00:45:33,440 It takes some time to do it the original proof by. 584 00:45:34,639 --> 00:45:35,420 Bowman. 585 00:45:37,510 --> 00:45:40,410 Was really quite clunky I mean he gave. 586 00:45:41,450 --> 00:45:46,070 Really wide error bounds and but they were they were big but 587 00:45:46,070 --> 00:45:51,050 they weren't too big so they kept getting bigger and bigger as you go along this T. 588 00:45:51,050 --> 00:45:52,530 gets bigger and remember T. 589 00:45:52,530 --> 00:45:53,780 is a function of N. 590 00:45:53,780 --> 00:45:58,359 so it was not enough to do it up to teak was ten so that values of T. 591 00:45:58,359 --> 00:46:02,080 you'd have an Arab and then here you'd have a wider Arab ound and 592 00:46:02,080 --> 00:46:06,719 somehow he set up these Arab ounces he said himself there was 593 00:46:06,719 --> 00:46:10,559 no nothing that they were the best ones but so 594 00:46:10,559 --> 00:46:15,410 if you're within here at this point you're within here at this point but 595 00:46:15,410 --> 00:46:18,530 then you have to do it when you get out to Epsilon Square event 596 00:46:18,530 --> 00:46:23,219 your arab ounds aren't so big that you've destroyed the whole process and 597 00:46:23,219 --> 00:46:28,190 he managed to do that so then once again I thought the story was over. 598 00:46:29,390 --> 00:46:30,760 But it wasn't over. 599 00:46:32,010 --> 00:46:37,130 And now we come to just the last couple of years 600 00:46:37,130 --> 00:46:42,250 two groups Tom Bowman again working with 601 00:46:42,250 --> 00:46:48,030 Peter key Vosh who is now it said Oxford I think yeah yeah 602 00:46:48,030 --> 00:46:52,840 it was a lot of wonderful results and what I like to call the empathy group. 603 00:46:54,280 --> 00:46:58,380 Because they were all it had in Rio de Janeiro. 604 00:47:00,340 --> 00:47:06,060 Ponte to Vero Gryphus and Morris and they looked at this process. 605 00:47:07,360 --> 00:47:13,560 Again and highly technical stuff but they managed to show in a certain way 606 00:47:13,560 --> 00:47:18,240 when you look at it in the right way you can get the differential equations to be 607 00:47:18,240 --> 00:47:23,299 self corrected so that is if they're off if you're off from the differential 608 00:47:23,299 --> 00:47:28,330 equation you're moving more you're tending to move back to the solution. 609 00:47:28,330 --> 00:47:33,770 Into the differential equations and because of that they could show that 610 00:47:33,770 --> 00:47:41,129 the solution stayed close to the differential equation longer then 611 00:47:41,129 --> 00:47:47,200 Bowman was able in fact they got it up to really right near the end so the end is 612 00:47:47,200 --> 00:47:52,130 when the number of open edges gets down to around and to the three halves. 613 00:47:54,089 --> 00:47:58,490 And there are reasons for that but but anyway that's the scaling so 614 00:47:58,490 --> 00:48:03,129 they really got it down to two with an end to the three have splits epsilon open 615 00:48:03,129 --> 00:48:08,469 edges which was really very close was effectively the end of the process 616 00:48:08,469 --> 00:48:13,010 and when they did that they were actually able to get 617 00:48:13,010 --> 00:48:16,830 reasonable constants the results of Cam and Boman just 618 00:48:16,830 --> 00:48:20,500 gave terrible constants they didn't really care they just wanted to get constants. 619 00:48:21,690 --> 00:48:29,410 But now we are and this is where we stand today that are of three K. 620 00:48:29,410 --> 00:48:33,159 is known within a factor of four. 621 00:48:34,189 --> 00:48:38,950 Or if you prefer if you make carry a function of N. 622 00:48:38,950 --> 00:48:41,640 it's known within a factor of two so it's even better. 623 00:48:43,880 --> 00:48:48,470 But this is tremendous progress. 624 00:48:49,790 --> 00:48:55,750 Let me and with an open question that I just really like. 625 00:48:57,280 --> 00:49:03,189 Take the triangle free graph all the way to the end all right so at the end 626 00:49:03,189 --> 00:49:08,060 you get a triangle free graph but furthermore it's a maximum triangle for 627 00:49:08,060 --> 00:49:13,619 a graph because every pair has been looked at OK so if you didn't put it in 628 00:49:13,619 --> 00:49:19,210 it means you couldn't put it in so every every pair was looked at and 629 00:49:19,210 --> 00:49:24,880 so the only way it wasn't put in is if there is a path of length two so 630 00:49:24,880 --> 00:49:29,358 any pair X.Y. 631 00:49:31,788 --> 00:49:35,398 that has not been put in it must be. 632 00:49:39,478 --> 00:49:40,798 The same no here 633 00:49:47,199 --> 00:49:50,759 there must be a path of length to tautologically. 634 00:49:51,789 --> 00:49:58,719 And I'll call a pair special if there is exactly one such pair 635 00:49:58,719 --> 00:50:04,259 and the question I have is what can you tell me about these special pairs 636 00:50:04,259 --> 00:50:09,948 I love to hear anything about them can you prove that there always are special 637 00:50:09,948 --> 00:50:15,289 pairs can you prove that they always that always in the probable istic sense 638 00:50:15,289 --> 00:50:20,159 that there always aren't special pairs can you say how many special 639 00:50:20,159 --> 00:50:25,228 pairs you get you say something about the structure of the special pairs we 640 00:50:25,228 --> 00:50:30,418 really don't know anything at all about that and. 641 00:50:31,709 --> 00:50:36,819 So it's nice to know I usually have a nice quote at the end but 642 00:50:36,819 --> 00:50:38,508 I forgot what I put on so. 643 00:50:40,319 --> 00:50:45,899 You know you go no I just here's another I had a phantom quote for the colloquium. 644 00:50:48,319 --> 00:50:52,569 So this is a nice one about doing mathematics 645 00:50:55,379 --> 00:50:56,709 and that's it thank you very much. 646 00:51:06,879 --> 00:51:07,379 Dana. 647 00:51:10,719 --> 00:51:16,108 Anyway why yes where was my impetus for 648 00:51:16,108 --> 00:51:21,119 it was I wanted to look from a logic point of view 649 00:51:21,119 --> 00:51:26,408 if you have a notion of a random structure and so the random triangle free 650 00:51:26,408 --> 00:51:30,259 I mean it's a distribution I told you the algorithm and 651 00:51:30,259 --> 00:51:34,529 what I wanted to say was I wanted to look at zero one laws and 652 00:51:34,529 --> 00:51:40,039 I thought there was going to be a zero one zero but the statement that there exists 653 00:51:40,039 --> 00:51:45,219 a special pair is a first order statement there exists there exists X. 654 00:51:45,219 --> 00:51:47,989 or Y. such that there's a unique that are not 655 00:51:47,989 --> 00:51:53,108 an edge that are not adjacent and that there's exactly one Z. 656 00:51:53,108 --> 00:51:56,889 That's a first order statement so that was my impetus for 657 00:51:56,889 --> 00:52:00,619 it and now I'm just intrigued by and but 658 00:52:00,619 --> 00:52:06,009 I I I I somehow my feeling is that 659 00:52:06,009 --> 00:52:11,189 all this tremendous work that's been done so far will not be particularly helpful 660 00:52:11,189 --> 00:52:16,159 here because the the I think of the special pairs as. 661 00:52:17,868 --> 00:52:22,949 The death throes of the of the process that is the process is really 662 00:52:22,949 --> 00:52:27,889 nearing the end you've only got a few more edges that you can possibly put 663 00:52:27,889 --> 00:52:32,639 in and you but I don't know and so somehow. 664 00:52:33,648 --> 00:52:39,109 My gut feeling is that all this beautiful work that has been done will not be 665 00:52:39,109 --> 00:52:46,548 helpful hopefully I'm yeah standing yeah it's emerging. 666 00:52:49,769 --> 00:52:52,989 Me see yes so you know as. 667 00:52:54,618 --> 00:52:59,449 You know this is a beautiful question this is a beautiful question so the. 668 00:53:00,819 --> 00:53:05,468 Shannon is the founder of information theory I call that. 669 00:53:07,019 --> 00:53:07,648 And. 670 00:53:08,938 --> 00:53:14,048 And the timing was just about the time of erudition but 671 00:53:14,048 --> 00:53:18,108 as near as I can tell and I was at Bell Labs at the time I mean so 672 00:53:18,108 --> 00:53:23,399 Shannon was the great God you know there but for some reason the air 673 00:53:23,399 --> 00:53:28,399 dish type work and all the people information theory at that time 674 00:53:28,399 --> 00:53:33,439 the they didn't connect you know we think of mathematics as unified but 675 00:53:33,439 --> 00:53:37,899 sometimes they're stories where there are different groups doing very similar things 676 00:53:37,899 --> 00:53:42,779 that somehow a don't get connected and let me add another example of that so so 677 00:53:42,779 --> 00:53:47,738 my answer is No I I mean I don't know but I don't think so but let me give another 678 00:53:47,738 --> 00:53:52,988 example which is the giant component that many of you are aware of. 679 00:53:54,029 --> 00:53:58,859 Again going back to to the mathematical physicists and 680 00:53:58,859 --> 00:54:04,079 the giant component is very directly connected it's sort of a discrete and 681 00:54:04,079 --> 00:54:07,999 I say an asymptotic analogue to percolation. 682 00:54:09,979 --> 00:54:14,609 That people in mathematical physics have worked on and now I mean I just came 683 00:54:14,609 --> 00:54:18,249 back from a meeting in Holland and they're the people in mathematical physics in 684 00:54:18,249 --> 00:54:23,018 the people in random graphs that are actually talking to each other but 685 00:54:23,018 --> 00:54:27,509 it's really quite remarkable that there were quite a few years where these two 686 00:54:27,509 --> 00:54:33,479 groups were working and they really didn't talk to each other and. 687 00:54:34,939 --> 00:54:38,609 I somehow I think this unfortunately happens you know the good 688 00:54:38,609 --> 00:54:43,679 stories are when people do talk to each other somehow I think these are two cases. 689 00:54:43,679 --> 00:54:47,118 Or they didn't it's really remarkable that that the Shannon 690 00:54:47,118 --> 00:54:51,738 group a move which I knew a lot of the people there and WEINER And 691 00:54:51,738 --> 00:54:56,138 Burleigh camps Lepi and all these great GILBERT You know but. 692 00:54:57,438 --> 00:55:02,319 But but they were doing this and heritage was doing random good and 693 00:55:02,319 --> 00:55:06,649 somehow they they just didn't make the connection that they were doing very 694 00:55:06,649 --> 00:55:09,768 similar stuff yeah that's it yeah 695 00:55:13,438 --> 00:55:18,288 I think you he was really struck by your remark about the challenge of finding 696 00:55:18,288 --> 00:55:23,689 the rights yes and papers like yes and 697 00:55:23,689 --> 00:55:28,199 it just made me think a little bit about what what you feel which is on I don't 698 00:55:30,308 --> 00:55:35,899 know you're going to play a long time on the United Nations and 699 00:55:35,899 --> 00:55:40,079 had to guess correctly combinations of variables you know. 700 00:55:41,439 --> 00:55:43,048 Yeah and then later on. 701 00:55:44,259 --> 00:55:45,739 Building is a. 702 00:55:46,778 --> 00:55:51,759 Very beautiful realize Asian ideas will build computers 703 00:55:51,759 --> 00:55:55,448 Yes and everything about what you said that 704 00:55:55,448 --> 00:56:00,419 day felt very much like you know one station Yes hundred resisted out rhythms. 705 00:56:05,410 --> 00:56:10,160 Yes there well to to be hope they will bank you very much 706 00:56:10,160 --> 00:56:14,860 I mean to give an amusing answer I think the algorithm is hard work and 707 00:56:14,860 --> 00:56:19,150 in the sense that you have to try a lot of things that fail and 708 00:56:19,150 --> 00:56:22,280 I think this scaling coming up with the right scaling. 709 00:56:24,080 --> 00:56:27,280 Is really critical but that that doesn't mean it's easy and 710 00:56:27,280 --> 00:56:32,150 when you find it everything fits nicely in certainly in my own experience. 711 00:56:33,769 --> 00:56:36,590 You know the papers may look nice but 712 00:56:36,590 --> 00:56:39,610 you know you don't see in the published paper all the things with the wrong 713 00:56:39,610 --> 00:56:44,780 scaling which have all these crazy factors floating around in the end they cancel but 714 00:56:44,780 --> 00:56:49,699 then when you do it with the right scaling so I agree it's very important but 715 00:56:49,699 --> 00:56:53,570 the only algorithm I know for finding the right scaling is is hard work or 716 00:56:53,570 --> 00:56:56,649 let me say it another way a lot of practice so 717 00:56:56,649 --> 00:57:01,030 I'll just tell a quick air dish story when I first met erudition you know would. 718 00:57:02,050 --> 00:57:06,939 Would would would be talking about a problem and have a pad of paper and 719 00:57:06,939 --> 00:57:12,750 you know some some problem and he said well let P. 720 00:57:12,750 --> 00:57:18,150 equal log log in over and 721 00:57:18,150 --> 00:57:22,000 to the three halves and then he'd go on and solve the problem. 722 00:57:25,189 --> 00:57:33,259 Walk where you got paid come from well you know now maybe I do that to my studio. 723 00:57:35,479 --> 00:57:37,189 Yeah but so that you for 724 00:57:37,189 --> 00:57:42,029 the commented about I mean I don't I don't I really don't think it's an algorithm for 725 00:57:42,029 --> 00:57:46,519 it I mean I don't think it's a definite procedure but I think that when you find 726 00:57:46,519 --> 00:57:51,428 the right scale and then you really get insight into the process you know. 727 00:57:53,279 --> 00:57:54,699 People felt the same way about. 728 00:57:55,939 --> 00:57:58,148 You know only right. 729 00:58:01,737 --> 00:58:02,237 And just. 730 00:58:03,878 --> 00:58:04,378 The more. 731 00:58:06,207 --> 00:58:06,878 I understand. 732 00:58:13,768 --> 00:58:18,817 Why this is critical that when you're looking here it's sort of equivalent 733 00:58:18,817 --> 00:58:24,678 in some sense to to Z D Were deal is extremely large or 734 00:58:24,678 --> 00:58:25,847 there's no locality and. 735 00:58:30,048 --> 00:58:31,678 Yeah you know I'm. 736 00:58:36,187 --> 00:58:43,018 Right then they're very awkward they're very awkward and indeed 737 00:58:43,018 --> 00:58:49,679 the complete graph is I won't say it's easy but it's doable in the sense 738 00:58:49,679 --> 00:58:54,199 just because of exactly what you're saying because we really are saying yes. 739 00:58:57,999 --> 00:58:58,649 Probably love. 740 00:59:00,298 --> 00:59:02,029 No no no you know. 741 00:59:03,249 --> 00:59:05,739 How let me count the number of tons of insulted you. 742 00:59:07,379 --> 00:59:07,879 Want to. 743 00:59:10,889 --> 00:59:16,589 OK Let me let let Tom in and I'll get used to yeah. 744 00:59:18,449 --> 00:59:19,859 Yeah. 745 00:59:19,859 --> 00:59:20,359 Yeah. 746 00:59:21,459 --> 00:59:21,959 Yeah. 747 00:59:23,048 --> 00:59:23,548 Yeah. 748 00:59:25,709 --> 00:59:26,209 Yeah. 749 00:59:32,128 --> 00:59:40,828 Yeah yeah yeah. 750 00:59:49,740 --> 00:59:50,399 So you go back to. 751 01:00:04,439 --> 01:00:08,259 That's great that's actually a new I mean that's a new era story I've never 752 01:00:08,259 --> 01:00:11,918 heard that one that's better great that's great you had to live it. 753 01:00:14,888 --> 01:00:15,388 Was. 754 01:00:18,808 --> 01:00:19,689 Screwing up. 755 01:00:21,218 --> 01:00:25,069 OK yeah so you know the difference in question but 756 01:00:25,069 --> 01:00:27,029 yeah the idea is you have this. 757 01:00:28,179 --> 01:00:31,959 Yeah you can think of that there's a kind of process right you do that a scaling and 758 01:00:31,959 --> 01:00:36,199 you get a deterministic function which is describe it actually Chris and 759 01:00:36,199 --> 01:00:41,729 you study Yeah well the deterministic function is with expectation so the thing 760 01:00:41,729 --> 01:00:46,648 is that the actual random process has fluctuations but but go ahead yes. 761 01:00:47,898 --> 01:00:49,968 I do carry Yeah I think you're. 762 01:00:50,988 --> 01:00:52,129 Essentially the same way. 763 01:00:53,139 --> 01:00:56,508 Through the limits of the scale of Yes I mean so 764 01:00:56,508 --> 01:00:57,869 what's the connection between like the. 765 01:01:00,910 --> 01:01:01,450 Diffusion. 766 01:01:02,670 --> 01:01:04,540 Mean for limited at this. 767 01:01:05,560 --> 01:01:10,589 Size so I don't know I don't know but let me put it 768 01:01:10,589 --> 01:01:15,100 this way your your area is more developed but also part of it is this thing of T. 769 01:01:15,100 --> 01:01:19,920 going to infinity which is I think this is really a different 770 01:01:19,920 --> 01:01:25,460 things that are interested in steady state in which we have to look to go to infinity 771 01:01:25,460 --> 01:01:30,470 that we have an interchange of limitation Maybe I should say ti beyond infinity 772 01:01:30,470 --> 01:01:36,870 you see because here we're looking at equals epsilon root law again so so T. 773 01:01:36,870 --> 01:01:41,680 is not just go see it's one thing to say what are the awesome topics as T. 774 01:01:41,680 --> 01:01:46,320 goes to infinity it's another thing to say you've got this parameter N. 775 01:01:46,320 --> 01:01:47,039 let T. 776 01:01:47,039 --> 01:01:52,300 will absolutely epsilon squared of log and that to me really adds but 777 01:01:52,300 --> 01:01:56,359 I think there are a lot of connections in the methodology to C.S. Yes there is 778 01:01:56,359 --> 01:02:02,580 something that people look at his joints Yeah and they look at something of the. 779 01:02:04,170 --> 01:02:08,320 One it was you know if someone is going to put it up and 780 01:02:08,320 --> 01:02:09,530 that's interesting maybe it's. 781 01:02:10,810 --> 01:02:15,110 This case yes well this is this this is very good. 782 01:02:17,040 --> 01:02:19,500 OK All right thank you very.