Combinatorics in Hungary and Extremal Set Theory

Transcription

1 Combinatorics in Hungary and Extremal Set Theory Gyula O.H. Katona Rényi Institute, Budapest Jiao Tong University Colloquium Talk October 22, 2014

2 Combinatorics in Hungary A little history. 1

3 Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. 2

4 Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. 3

5 Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1948, one and a half year long war. 4

6 Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1948, one and a half year long war. Hungary was winning, Austria asked the help of the Russians. The two big countries easily suppressed the revolution. 5

7 Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. The united European armies pushed out the Turks around 1670, Hungary became a colony of Austria. Revolution and freedom fight against Austria in 1848, one and a half year long war. Hungary was winning, Austria asked the help of the Russians. The two big countries easily suppressed the revolution. Agreement in The Austrian Monarchy became the Austro- Hungarian Monarchy. Very fast economic progress in the Hungarian part. 6

8 Combinatorics in Hungary Educational reforms on every level. 7

9 Combinatorics in Hungary Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula König. 8

10 Combinatorics in Hungary Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula König. The world very first mathematical journal for high school students: 1896! 9

11 Combinatorics in Hungary Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula König. The world very first mathematical journal for high school students: 1896! The first mathematical contest for high school students in the world in

12 Combinatorics in Hungary Educational reforms on every level. General level of elementary schools raised. Talent search. Good math professor at universities. One of them: Gyula König. The world very first mathematical journal for high school students: 1894! The first mathematical contest for high school students in the world in 1894! Harsányi (NP), Von Neumann, Teller, Wigner (NP) came from the same high school in Budapest. Szilárd went to another strong school. 11

13 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. 12

14 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. Can all the girls find husbands? 13

15 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. Necessary condition: every set A of girls need to know at least A boys. 14

16 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. Necessary condition: every set A of girls need to know at least A boys. 15

17 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. Necessary condition: every set A of girls need to know at least A boys. 16

18 Marriage problem = Perfect Matching Given n girls and n boys, their knowing each other is given with a bipartite graph. Theorem (Dénes König, 1916, Hall, 1935) The condition is necessary and sufficient. 17

19 First book in Graph Theory by Dénes König,

20 First book in Graph Theory by Dénes König, Under his influence Erdős, Gallai, Szekeres, Turán started to think about graphs. 19

21 First book in Graph Theory by Dénes König, Under his influence Erdős, Gallai, Szekeres, Turán started to think about graphs. Erdős and Szekeres found the finite version of Ramsey theorem. 20

22 First book in Graph Theory by Dénes König, Under his influence Erdős, Gallai, Szekeres, Turán started to think about graphs. Erdős and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. 21

23 First book in Graph Theory by Dénes König, Under his influence Erdős, Gallai, Szekeres, Turán started to think about graphs. Erdős and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erdős went to England and the USA. 22

24 First book in Graph Theory by Dénes König, Under his influence Erdős, Gallai, Szekeres, Turán started to think about graphs. Erdős and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erdős went to England and the USA. Turán was in a labor camp, yet he was doing graph theory there! 23

25 Erdős, Ko, Rado Notation: [n] = {1, 2,..., n}. A family F 2 [n] is intersecting if F G holds for every pair F, G F. 24

26 Erdős, Ko, Rado Notation: [n] = {1, 2,..., n}. A family F 2 [n] is intersecting if F G holds for every pair F, G F. 25

27 Erdős, Ko, Rado ( Theorem (Erdős Ko Rado, done in 1938, published in 1961) If F [n] ) k is intersecting where k n 2 then ( ) n 1 F. k 1 26

28 I found a short proof in Erdős said: it is from the BOOK. 27

29 Erdős, Ko, Rado I found a short proof in Erdős said: it is from the BOOK. Martin Aigner and Günter Ziegler: Proofs from THE BOOK. Springer

30 Erdős, Ko, Rado I found a short proof in Erdős said: it is from the BOOK. Martin Aigner and Günter Ziegler: Proofs from THE BOOK. Springer Erdős returned to Hungary in

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33 Alfréd Rényi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. 32

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35 Random graphs Alfréd Rényi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erdős and Rényi started to study RANDOM GRAPHS. 34

36 Random graphs Alfréd Rényi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erdős and Rényi started to study RANDOM GRAPHS. 3 different models: (1) One graph is chosen randomly from the possible (( n ) 2) e graphs with e edges. 35

37 (2) Choose the edges independently with probability p = e ( n 2). 36

38 Random graphs (3) Add edges one by one and look at the graph when e edges are added. They are equivalent for reasonable problems. 37

39 Random graphs Theorem The first cycle appears around e = n 2. Theorem The graph becomes connected around e = 1 2n log n. 38

40 Random graphs Theorem The first cycle appears around e = n 2. Theorem The graph becomes connected around e = 1 2n log n. Before that, giant component. 39

41 Random graphs Many applications in natural sciences. E.g. physics, social graphs, of course, with other probabilities. 40

42 Shadows Definition. The shadow σ(a) of A is the family of all k 1-element sets obtained from the members of A by deleting exactly one element. σ(a) = {B : B = k 1, B A A} 41

43 Shadows Definition. The shadow σ(a) of A is the family of all k 1-element sets obtained from the members of A by deleting exactly one element. σ(a) = {B : B = k 1, B A A} 42

44 Shadows Definition. The shadow σ(a) of A is the family of all k 1-element sets obtained from the members of A by deleting exactly one element. σ(a) = {B : B = k 1, B A A} 43

45 Shadows Definition. The shadow σ(a) of A is the family of all k 1-element sets obtained from the members of A by deleting exactly one element. σ(a) = {B : B = k 1, B A A} 44

46 Shadows Definition. The shadow σ(a) of A is the family of all k 1-element sets obtained from the members of A by deleting exactly one element. σ(a) = {B : B = k 1, B A A} 45

47 Shadows Given n, k and F, minimize σ(f). If lucky then F = ( a k) holds for an integer a then the best construction is min σ(f) = ( ) a k 1 46

48 otherwise Lemma If 0 < k, m are integers then one can find integers a k > a k 1 >... > a t t 1 such that m = ( ak k ) + ( ) ak 1 k ( at t ) and they are unique. This is called the canonical form of m. 47

49 otherwise Shadow Theorem (Kruskal-K, 1960 s) If n, k and F are given the canonical form of F is F = ( ak k ) + ( ) ak 1 k ( at t ) then min σ(f) = ( ) ak k 1 + ( ) ak 1 k ( ) at. t 1 48

50 Szemerédi s regularity lemma 49

51 Szemerédi s regularity lemma Large graphs follow some pattern, some rules. 50

52 Szemerédi s regularity lemma 51

53 Szemerédi s regularity lemma 52

54 Szemerédi s regularity lemma 53

55 Szemerédi s regularity lemma Regularity lemma(szemerédi, 1975) A graph with many (many-many) vertices can be partitioned into equally sized subsets in such a way that most pairs of subsets span an ε-regular bipartite graph. 54

56 A problem motivated by cryptology A, B ( ) [n], A = B 3 σ(a) σ(b) = (In other words, if A A, B B then A B 1.) Find f(n, 3) = max A under these conditions. 55

57 Only an estimate ( ) x If x is a real number, k = x(x 1)...(x k+1) k!. Theorem. (Lovász version of the Shadow theorem) If A is a family of k- element sets, then A = ( ) x k σ(a) ( ) x. k 56

58 A weak estimate from Shadow Theorem Choose x in this way: A = B = ( ) x 3 By the Shadow Theorem: ( x 2) σ(a), σ(b) ( ) x 2 σ(a) + σ(b) 2 ( ) n 2 From here, asymptotically A = ( ) x 3 1 ( ) n 2 (1 + o(1))

59 Trivial construction gives: A = n3 (1 + o(1)). 48 A = { (a, b, c) : a < b < c n }, B = 2 {(a, b, c) : n 2 a < b < c }. 58

60 A better construction: A = n3 24 (1 + o(1)). A = { (a, b, c) : b+c 2 } n 2 { B = {(a, b, c) : n 2 < } a+b 2. 59

61 We have 0.25 lim sup f(n,3) ( n 3)

62 We have 0.25 lim sup f(n,3) ( n 3) Theorem (Frankl-Kato-Katona-Tokushige) ( ) n f(n, 3) = (1 + o(1)) 3 61

63 Theorem (Frankl-Kato-Katona-Tokushige) f(n, 3) = κ 3 ( n 3 ) (1 + o(1)) where κ is the unique real root in the (0,1)-interval of the equation z 3 = (1 z) 3 + 3z(1 z) 2. 62

64 Theorem (Frankl-Kato-Katona-Tokushige, 2013) A 1, A 2 ( ) [n] k, A1 = A 2, and A 1 A 1, A 2 A 2 imply A 1 A 2 1 then max A = µ k k ( ) n (1 + o(1)) k where µ k is the unique real root in the (0,1)-interval of the equation z k = (1 z) k + kz(1 z) k 1. 63

65 Some months later Huang, Linial, Naves, Peled, Sudakov proved a very similar result. They considered two families A ( ) ( [n] k, B [n] ) l where A A, b B imlies A B 1. Supposing A ( n k) = α they asymptotically determine max B ( n l). The case k = l gives back our result, but their upper estimate is less sharp. 64

66 Some months later Huang, Linial, Naves, Peled, Sudakov proved a very similar result. They considered two families A ( ) ( [n] k, B [n] ) l where A A, b B imlies A B 1. Supposing A ( n k) = α they asymptotically determine max B ( n l). The case k = l gives back our result, but their upper estimate is less sharp. Their motivation is theoretical. 65

67 Badly needed generalizations s-shadows (s k 2), that is, A B r. 66

68 Badly needed generalizations s-shadows (s k 2), that is, A B r. More families rather than only 2. 67

69 A family F 2 [n] is called a (u, v)-union-intersecting if for different members F 1,..., F u, G 1,..., G v the following holds: ( u i=1f i ) ( v j=1g j ). 68

70 A family F 2 [n] is called a (u, v)-union-intersecting if for different members F 1,..., F u, G 1,..., G v the following holds: ( u i=1f i ) ( v j=1g j ). Theorem (Katona-D.T. Nagy 2014+) Let 1 u v and suppose that the family F ( ) [n] k is a (u, v)-union intersecting family then holds if n > n(k, v). F ( ) n 1 k 1 + u 1 69

71 A family F 2 [n] is called a (u, v)-union-intersecting if for different members F 1,..., F u, G 1,..., G v the following holds: ( u i=1f i ) ( v j=1g j ). Theorem (Katona-D.T. Nagy 2014+) Let 1 u v and suppose that the family F ( ) [n] k is a (u, v)-union intersecting family then holds if n > n(k, v). F ( ) n 1 k 1 + u 1 70

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