Extremal problems concerning tournaments Timothy Chan (Monash) Andrzej Grzesik (Krak´ ow) Dan Kr´ al’ (Brno and Warwick) Jon Noel (Warwick) May 19, 2019 1
Overview of the talk • Extremal problems in tournaments some old and less old results • Tur´ an type problems in graphs Tur´ an’s Theorem Erd˝ os-Rademacher problem • Tur´ an type problems in tournaments cycles of length three and four cycles of arbitrary lengths 2
Tournaments • tournament = orientation of a complete graph • two possible 3-vertex subgraphs: C 3 and T 3 • edge v i → v j for i < j with probability p ∈ [1 / 2 , 1] � n • the number of C 3 is between 0 and 1 � + O ( n 2 ) 4 3 • the number of paths u → v → w is at most n 3 / 4 each C 3 contains 3 such paths, each T 3 one the number of C 3 is at most n 3 / 4 − n 3 / 6 = n 3 24 + O ( n 2 ) 2 3
Quasirandom tournaments • When does a tournament look random? random tournament = orient each edge randomly • When does a graph look random? • Thomason, and Chung, Graham and Wilson (1980’s) density of K 2 is p , density of C 4 is p 4 equivalent subgraph density conditions equivalent uniform density conditions equivalent spectral conditions . . . 4
Quasirandom tournaments • When does a tournament look random? • Coregliano, Razborov (2017) density of T 4 is 4! / 2 6 (unique minimizer) density of T k is k ! / 2( k 2 ) for k ≥ 4 • Other tournaments forcing quasirandom? Coregliano, Parente, Sato (2019) unique maximizer of a 5-vertex tournament 5
Overview of the talk • Extremal problems in tournaments some old and less old results • Tur´ an type problems in graphs Tur´ an’s Theorem Erd˝ os-Rademacher problem • Tur´ an type problems in tournaments cycles of length three and four cycles of arbitrary lengths 6
Tur´ an problems • Maximum edge-density of H -free graph 1 • Mantel’s Theorem (1907): 2 for H = K 3 ( K n 2 ) 2 , n ℓ − 2 • Tur´ an’s Theorem (1941): ℓ − 1 for H = K ℓ ( K ℓ − 1 ) n n ℓ − 1 ,..., χ ( H ) − 2 • Erd˝ os-Stone Theorem (1946): χ ( H ) − 1 • extremal examples unique up to o ( n 2 ) edges 7
Erd˝ os-Rademacher problem • Tur´ an’s Theorem: edge-density ≤ 1 / 2 ⇔ minimum triangle density = 0 • What happens if edge-density > 1 / 2? • minimum attained by K n,...,n for edge-density k − 1 k • smooth transformation from K n,n for K n,n,n , from K n,n,n to K n,n,n,n , etc. 8
Erd˝ os-Rademacher problem K 3 1 0 K 2 1 9
History of the problem • Goodman bound (1959) d ( K 3 , G ) ≥ 2 d ( K 2 , G ) × ( d ( K 2 , G ) − 1 / 2 ) true for d ( K 2 , G ) = k − 1 k • Bollob´ as (1976) contained in the convex hull “linear” approximation • Lov´ asz and Simonovits (1983) � k − 1 k , k − 1 � true for d ( K 2 , G ) ∈ + ε k k • Fisher (1989) true for d ( K 2 , G ) ∈ [1 / 2 , 2 / 3] 10
Solution of the problem • solved by Razborov in 2008 • Flag Algebra Method calculus for subgraph densities multiplication of linear combinations search for true inequalities using SDP • additional proof idea differential method (local modifications) 11
Extensions • Nikiforov (2011) minimum density of K 4 • Reiher (2016) minimum density of K r • Pikhurko and Razborov (2017) asymptotic structure of extremal graphs • Liu, Pikhurko and Staden (2017+) exact structure of extremal graphs 12
Structure of extremal graphs • Pikhurko and Razborov (2017) asymptotic structure of extremal graphs • extremal graphs K n,...,n,αn K n,αn → triangle-free graph on (1 + α ) n vertices • no K 1 , 2 = K 1 ∪ K 2 ⇒ K n,...,n,αn only 13
Overview of the talk • Extremal problems in tournaments some old and less old results • Tur´ an type problems in graphs Tur´ an’s Theorem Erd˝ os-Rademacher problem • Tur´ an type problems in tournaments cycles of length three and four cycles of arbitrary lengths 14
Tournaments • tournament: density parameterized by C 3 • analogue of Erd˝ os-Rademacher Problem minimum density of C 4 for a fixed density of C 3 • Conjecture of Linial and Morgenstern (2014) blow-up of a transitive tournament (random inside) with all but one equal parts and a smaller part transitive orientation of K n,...,n,αn , random inside parts 15
Tournaments • minimum density of C 4 for a fixed density of C 3 • Conjecture of Linial and Morgenstern (2014) blow-up of a transitive tournament (random inside) with all but one equal parts and a smaller part 16
Our results t ( C 4 , T ) 1 12 1 16 1 128 1 432 0 t ( C 3 , T ) 1 1 1 72 32 8 17
Approach to the problem • linear algebra tools adjacency matrix A ∈ { 0 , 1 } V ( G ) × V ( G ) Tr A k = number of closed k -walks • regularity method approximation by an ( n × n )-matrix A rows and columns ≈ parts in regularity decomposition A ij ≥ 0 and A ij + A ji = 1 for all i, j ∈ { 1 , . . . , n } 18
Cases of two and three parts • non-negative matrix A , s.t. A + A T = J • properties of the spectrum of A : Tr A = λ 1 + . . . + λ k = 1 / 2 Perron–Frobenius ⇒ ∃ ρ ∈ R : ρ = λ 1 and | λ i | ≤ λ 1 v ∗ ( A + A T ) v = v ∗ ( λ i + λ i ) v = v ∗ J v ≥ 0 ⇒ Re λ i ≥ 0 • fix Tr A 3 = λ 3 1 + . . . + λ 3 k ∈ [1 / 36 , 1 / 8] minimize Tr A 4 = λ 4 1 + . . . + λ 4 k • optimum λ ≤ k − 1 = ρ and λ k = 1 / 2 − ( k − 1) ρ , k ∈ { 2 , 3 } 19
Case of two parts—structure • A = ( J + B ) / 2, B is antisymmetric, i.e. B = − B T A is non-negative and A + A T = J • analysis of antisymmetric matrix B i cos 2 α i = 1 σ i and α i for matrix B with � 0 σ 1 0 0 − σ 1 0 0 0 B = U T U 0 0 0 σ 2 0 0 − σ 2 0 20
Case of two parts—structure • A = ( J + B ) / 2, B is antisymmetric, i.e. B = − B T A is non-negative and A + A T = J • analysis of antisymmetric matrix B i cos 2 α i = 1 σ i and α i for matrix B with � • Tr A 3 ≈ Tr J 3 + Tr J B 2 = � i cos 2 α i i σ 2 Tr A 4 ≈ Tr J 4 + Tr J 2 B 2 + Tr B 4 ≈ Tr J B 2 + � i σ 4 i • optimum for α 1 = 0, α ≥ 2 = π/ 2 and σ ≥ 2 = 0 21
Case of two parts—structure • A = ( J + B ) / 2, B is antisymmetric, i.e. B = − B T i cos 2 α i = 1 σ i and α i for matrix B with � optimum for α 1 = 0, α ≥ 2 = π/ 2 and σ ≥ 2 = 0 • assign p v ∈ [0 , 1 / 2] to each vertex v orient from v to w with probability 1 / 2 + ( p v − p w ) • conjectured construction: p v ∈ { 0 , 1 / 2 } 22
Maximum density of cycles • work in progress with Grzesik, Lov´ asz Jr. and Volec • What is maximum density of cycles of length k ? k ≡ 1 mod 4 ⇔ regular tournament k ≡ 2 mod 4 ⇔ quasirandom tournament k ≡ 3 mod 4 ⇔ regular tournament k ≡ 4 mod 4 ⇔ ???? • “cyclic” tournament for k = 4 and k = 8 23
Thank you for your attention! 24
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