Combinatorics in Hungary and Extremal Set Theory Gyula O.H. Katona R´ enyi Institute, Budapest Jiao Tong University Colloquium Talk October 22, 2014
Combinatorics in Hungary A little history. 1
Combinatorics in Hungary A little history. Turkey occupied half of Hungary in 1526, Austria the other half. 2
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
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
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
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 1867. The Austrian Monarchy became the Austro- Hungarian Monarchy . Very fast economic progress in the Hungarian part. 6
Combinatorics in Hungary Educational reforms on every level. 7
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 K¨ onig. 8
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 K¨ onig. The world very first mathematical journal for high school students: 1896! 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 K¨ onig. The world very first mathematical journal for high school students: 1896! The first mathematical contest for high school students in the world in 1920. 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 K¨ onig. The world very first mathematical journal for high school students: 1894! The first mathematical contest for high school students in the world in 1894! Hars´ anyi (NP), Von Neumann, Teller, Wigner (NP) came from the same high school in Budapest. Szil´ ard went to another strong school. 11
Marriage problem = Perfect Matching Given n girls and n boys, their “knowing each other” is given with a bipartite graph. 12
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
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
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
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
Marriage problem = Perfect Matching Given n girls and n boys, their “knowing each other” is given with a bipartite graph. Theorem (D´ enes K¨ onig, 1916, Hall, 1935) The condition is necessary and sufficient. 17
First book in Graph Theory by D´ enes K¨ onig, 1936. 18
First book in Graph Theory by D´ enes K¨ onig, 1936. Under his influence Erd˝ os, Gallai, Szekeres, Tur´ an started to think about graphs. 19
First book in Graph Theory by D´ enes K¨ onig, 1936. Under his influence Erd˝ os, Gallai, Szekeres, Tur´ an started to think about graphs. Erd˝ os and Szekeres found the finite version of Ramsey theorem. 20
First book in Graph Theory by D´ enes K¨ onig, 1936. Under his influence Erd˝ os, Gallai, Szekeres, Tur´ an started to think about graphs. Erd˝ os and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. 21
First book in Graph Theory by D´ enes K¨ onig, 1936. Under his influence Erd˝ os, Gallai, Szekeres, Tur´ an started to think about graphs. Erd˝ os and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erd˝ os went to England and the USA. 22
First book in Graph Theory by D´ enes K¨ onig, 1936. Under his influence Erd˝ os, Gallai, Szekeres, Tur´ an started to think about graphs. Erd˝ os and Szekeres found the finite version of Ramsey theorem. Because of the nazism, Szekeres fled to Shanghai, then to Australia. Erd˝ os went to England and the USA. Tur´ an was in a labor camp, yet he was doing graph theory there! 23
Erd˝ os, 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
Erd˝ os, 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
Erd˝ os, Ko, Rado Theorem (Erd˝ os – Ko – Rado, done in 1938, published in 1961) If F ⊂ � [ n ] � is intersecting where k ≤ n 2 then k � n − 1 � |F| ≤ . k − 1 26
os said: it is from the BOOK . I found a short proof in 1972. Erd˝ 27
Erd˝ os, Ko, Rado os said: it is from the BOOK . I found a short proof in 1972. Erd˝ Martin Aigner and G¨ unter Ziegler: Proofs from THE BOOK. Springer 1998. 28
Erd˝ os, Ko, Rado os said: it is from the BOOK . I found a short proof in 1972. Erd˝ Martin Aigner and G¨ unter Ziegler: Proofs from THE BOOK. Springer 1998. Erd˝ os returned to Hungary in 1954. 29
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Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. 32
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Random graphs Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erd˝ os and R´ enyi started to study RANDOM GRAPHS. 34
Random graphs Alfr´ ed R´ enyi a strong probabilist, director of the Mathematical Institute of the Hungarian Academy of Sciences. Erd˝ os and R´ enyi 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
(2) Choose the edges independently with probability e p = � . � n 2 36
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
Random graphs Theorem The first cycle appears around e = n 2 . Theorem The graph becomes connected around e = 1 2 n log n . 38
Random graphs Theorem The first cycle appears around e = n 2 . Theorem The graph becomes connected around e = 1 2 n log n . Before that, “giant component”. 39
Random graphs Many applications in natural sciences. E.g. physics, social graphs, of course, with other probabilities. 40
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
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
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
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
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
Shadows Given n, k and |F| , minimize | σ ( F ) | . � a � If lucky then |F| = holds for an integer a then the best construction is k � a � min | σ ( F ) | = k − 1 46
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