Exact solution of the Erd os-S os conjecture Mikl os Ajtai J - PowerPoint PPT Presentation

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Exact solution of the Erd˝ os-S´ os conjecture Mikl´ os Ajtai J´ anos Koml´ os, Mikl´ os Simonovits, Endre Szemer´ edi Alfr´ ed R´ enyi Math Inst Budapest Nyborg, 2015

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Exact solution of the Erd˝ os-S´ os conjecture Mikl´ os Ajtai J´ anos Koml´ os, Mikl´ os Simonovits, Endre Szemer´ edi Alfr´ ed R´ enyi Math Inst Budapest Nyborg, 2015

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Starting right in the middle 2 N1: G n , P k , T k . Theorem (Ajtai Koml´ os Simonovits Szemer´ edi) There exists a k 0 such that for k > k 0 , for any tree T k , if e ( G n ) > 1 2( k − 2) n then T k ֒ → G n .

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Starting right in the middle 2 N1: G n , P k , T k . Theorem (Ajtai Koml´ os Simonovits Szemer´ edi) There exists a k 0 such that for k > k 0 , for any tree T k , if e ( G n ) > 1 2( k − 2) n then T k ֒ → G n . The general question: Given a sample graph L , how many edges can G n have, without containing L . N2: ex ( n , L ), = maximum number of edge . . . EX ( n , L ). The family of Extremal Graphs = G n attaining the maximum

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Tur´ an’s questions 3 Tur´ an was motivated (basically) by Ramsey ’s theorem Tur´ an asked the extremal number for various excluded subgraphs: cube, icosahedron, octahedron, dodecahedron, For us the important case is: path P k . and trees T k

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Erd˝ os-S´ os conjecture and its motivation 4 Kk− K Kk− 1 k− 1 ... 1 Kr Figure : Z n , k Theorem (Erd˝ os-Gallai) ex ( n , P k ) ≤ 1 2( k − 2) n . The extremal graph is Z n , k . (!) If S k is the star, then, trivially, ex ( n , S k ) ≤ 1 2( k − 2) n .

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Erd˝ os-S´ os conjecture 5 For any T k , ex ( n , T k ) ≤ 1 2( k − 2) n . In other words, If e ( G n ) > 1 2( k − 2) n , then G n contains each k -vertex tree. Easy: ex ( n , T k ) ≤ ( k − 2) n . Kk− K Kk− 1 k− 1 ... 1 Kr

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases The conjecture/ other ≈ -extremal structure 6 For any fixed tree T k , ex ( n , T k ) ≤ 1 2( k − 2) n .

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases The conjecture/ other ≈ -extremal structure 6 For any fixed tree T k , ex ( n , T k ) ≤ 1 2( k − 2) n . k−1 2 Kk− K Kk− 1 k− 1 ... 1 k−1 n− 2 Kr Z n , k W n ,κ

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Motivation 7 Claim (Folklore) If d min ( G n ) ≥ k − 1 , then T k ֒ → G n , for every tree T k . Greedy embedding True for stars S k : Trivial True for paths P k : Erd˝ os-Gallai. It would be trivial, if G n were regular! What is the difficulty? That the vertices of G n may have (very) different degrees, and em- bedding T k step by step, we may arrive at a vertex g ∈ T k having large degree, and when we try to put it down into x ∈ G n , all its neighbours are already used up.

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Some known cases 8 Sidorenko, if there is an x ∈ V ( G n ) with n / 2 leaves Dobson, it the girth is “large” Brandt-Dobson Wozniak Theorem (McLennan) If diameter ( T k ) ≤ 4 , then ES Conjecture holds. . . .

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases The Loebl-Koml´ os-S´ os conjecture 9 Conjecture (Loebl–Koml´ os–S´ os Conjecture 1995) Suppose that G is an n-vertex graph with at least n / 2 vertices of degree more than k − 2 . Then G contains each tree of order k. k 2 − 1 k 2 − 1 k 2 − 1 k 2 + 1 k 2 + 1 k 2 + 1

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Motivation? 10 Erd˝ os-F¨ uredi-Loebl-S´ os: Uniform distribution for graphs Ramsey for moncohromatic trees They needed the simplest form of this conjecture: The Loebl Conjecture (i.e. n = k ). Koml´ os and S´ os generalized the Loebl conjecture. For paths there were already several similar results: Woodall Erd˝ os-Faudree-Schelp-Simonovits results on the Ramsey numbers of a fixed graph versus a large tree. Hao Li ...

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What happend? 11 Ajtai-Koml´ os-Szemer´ edi: Proof of the Approximative weakening of the Loebl Conjecture. Yi Zhao: Exact solution for large k . Piguet-Stein / Oliver Cooley: a big step forward. Piguet-Hladk´ y

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Details? 12 Conjecture (Weaker, approximate version) If at least 1 2 (1 + η ) n vertices of G n have degree at least (1 + η ) k, then T k ֒ → G n . Ajtai-Koml´ os-Szemer´ edi Yi Zhao Piguet-Stein / Cooley Theorem (Hladky-Koml´ os-Piguet-Simonovits-Stein-Szemer´ edi) os Conjecture holds for k > k 0 . The Koml´ os-S´ Arxiv ( > 160pp) + Short description + three out of four papers accepted for publications

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Why is this problem difficult? II 13 Uniqueness of extremal graphs Those problems are easy, where there is a main property of the (conjectured) extremal graphs “governing” the proof. Here there are two (almost) extremal graphs, of completely different structures. Many graphs G n many different trees T k

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Plan? 14 Is it easy for generalized random graphs? If YES, then Regularity Lemma may help. What is a Generalized Random Graph ? What is the Regularity Lemma Why and when does the Regularity Lemma help? Does it help NOW?

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What is a Generalized Random graph? 15 A matrix A = ( p ij ) r × r of probabilities is given. We divide n vertices into r classes U i and join each x ∈ U i to y ∈ U j independently, with probability p ij d ( X , Y ) := e ( X , Y ) | X || Y | Definition ( ε -regular pair ( A , B ) in G n ) ... if whenever X ⊆ A and | X | > ε | A | and | Y | > ε | B | , then | d ( X , Y ) − d ( A , B ) | < ε.

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases Important “test”: Generalized Random Graphs 16 If we can solve an extremal graph problem “easily” for Generalized Random Graphs, then we probably can also solve it for any dense graphs sequence.

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What is the Regularity Lemma? 17 Informally: Each graph can be approximated by generalized random graphs / generalized quasi-random graphs

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What is the Regularity Lemma? 17 Informally: Each graph can be approximated by generalized random graphs / generalized quasi-random graphs Theorem (Szemer´ edi Regularity Lemma) For every ε > 0 and ν 0 there exists a ν 1 ( ε, ν 0 ) such that for every G n , V ( G n ) can be partitioned into ν sets U 1 , . . . , U ν , for some ν 0 < ν < ν 1 ( ε, ν 0 ) , so that || U i | − | U j || ≤ 1 for every i , j > 0 , and � ν � U i U j is ε –regular for all but at most ε pairs ( i , j ) . 2

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What is the Regularity Lemma? 17 Informally: Each graph can be approximated by generalized random graphs / generalized quasi-random graphs Theorem (Szemer´ edi Regularity Lemma) For every ε > 0 and ν 0 there exists a ν 1 ( ε, ν 0 ) such that for every G n , V ( G n ) can be partitioned into ν sets U 1 , . . . , U ν , for some ν 0 < ν < ν 1 ( ε, ν 0 ) , so that || U i | − | U j || ≤ 1 for every i , j > 0 , and � ν � U i U j is ε –regular for all but at most ε pairs ( i , j ) . 2 Cluster graph

The conjecture and its motivation Dense case, and why is it easier? Sharp and Approximate cases What is the Regularity Lemma? 17 Informally: Each graph can be approximated by generalized random graphs / generalized quasi-random graphs Theorem (Szemer´ edi Regularity Lemma) For every ε > 0 and ν 0 there exists a ν 1 ( ε, ν 0 ) such that for every G n , V ( G n ) can be partitioned into ν sets U 1 , . . . , U ν , for some ν 0 < ν < ν 1 ( ε, ν 0 ) , so that || U i | − | U j || ≤ 1 for every i , j > 0 , and � ν � U i U j is ε –regular for all but at most ε pairs ( i , j ) . 2 Cluster graph