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Extremal problems concerning tournaments Timothy Chan (Monash) - - PowerPoint PPT Presentation

May 12, 2023 176 likes •430 views

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

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Extremal problems concerning tournaments

Timothy Chan (Monash) Andrzej Grzesik (Krak´

  • w)

Dan Kr´ al’ (Brno and Warwick) Jon Noel (Warwick) May 19, 2019

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SLIDE 2

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˝

  • s-Rademacher problem
  • Tur´

an type problems in tournaments cycles of length three and four cycles of arbitrary lengths

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Tournaments

  • tournament = orientation of a complete graph
  • two possible 3-vertex subgraphs: C3 and T3
  • edge vi → vj for i < j with probability p ∈ [1/2, 1]
  • the number of C3 is between 0 and 1

4

n

3

  • + O(n2)
  • the number of paths u → v → w is at most n3/4

each C3 contains 3 such paths, each T3 one the number of C3 is at most n3/4−n3/6

2

= n3

24 + O(n2) 3

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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 K2 is p, density of C4 is p4 equivalent subgraph density conditions equivalent uniform density conditions equivalent spectral conditions . . .

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Quasirandom tournaments

  • When does a tournament look random?
  • Coregliano, Razborov (2017)

density of T4 is 4!/26(unique minimizer) density of Tk is k!/2(

k 2) for k ≥ 4

  • Other tournaments forcing quasirandom?

Coregliano, Parente, Sato (2019) unique maximizer of a 5-vertex tournament

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SLIDE 6

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˝

  • s-Rademacher problem
  • Tur´

an type problems in tournaments cycles of length three and four cycles of arbitrary lengths

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Tur´ an problems

  • Maximum edge-density of H-free graph
  • Mantel’s Theorem (1907):

1 2 for H = K3 (K n

2 , n 2 )

  • Tur´

an’s Theorem (1941):

ℓ−2 ℓ−1 for H = Kℓ (K

n ℓ−1 ,..., n ℓ−1 )

  • Erd˝
  • s-Stone Theorem (1946):

χ(H)−2 χ(H)−1

  • extremal examples unique up to o(n2) edges

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Erd˝

  • s-Rademacher problem
  • Tur´

an’s Theorem: edge-density ≤ 1/2 ⇔ minimum triangle density = 0

  • What happens if edge-density > 1/2?
  • minimum attained by Kn,...,n for edge-density k−1

k

  • smooth transformation from Kn,n for Kn,n,n,

from Kn,n,n to Kn,n,n,n, etc.

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Erd˝

  • s-Rademacher problem

K2 1 K3 1

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History of the problem

  • Goodman bound (1959)

d(K3, G) ≥ 2d(K2, G) × ( d(K2, G) − 1/2 ) true for d(K2, G) = k−1

k

  • Bollob´

as (1976) contained in the convex hull “linear” approximation

  • Lov´

asz and Simonovits (1983) true for d(K2, G) ∈ k−1

k , k−1 k

+ εk

  • Fisher (1989)

true for d(K2, G) ∈ [1/2, 2/3]

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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)

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Extensions

  • Nikiforov (2011)

minimum density of K4

  • Reiher (2016)

minimum density of Kr

  • Pikhurko and Razborov (2017)

asymptotic structure of extremal graphs

  • Liu, Pikhurko and Staden (2017+)

exact structure of extremal graphs

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Structure of extremal graphs

  • Pikhurko and Razborov (2017)

asymptotic structure of extremal graphs

  • extremal graphs Kn,...,n,αn

Kn,αn → triangle-free graph on (1 + α)n vertices

  • no K1,2 = K1 ∪ K2 ⇒ Kn,...,n,αn only

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SLIDE 14

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˝

  • s-Rademacher problem
  • Tur´

an type problems in tournaments cycles of length three and four cycles of arbitrary lengths

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Tournaments

  • tournament: density parameterized by C3
  • analogue of Erd˝
  • s-Rademacher Problem

minimum density of C4 for a fixed density of C3

  • 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 Kn,...,n,αn, random inside parts

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Tournaments

  • minimum density of C4 for a fixed density of C3
  • Conjecture of Linial and Morgenstern (2014)

blow-up of a transitive tournament (random inside) with all but one equal parts and a smaller part

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Our results

t(C3, T) t(C4, T)

1 8 1 32 1 72 1 12 1 16 1 128 1 432

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Approach to the problem

  • linear algebra tools

adjacency matrix A ∈ {0, 1}V (G)×V (G) Tr Ak = number of closed k-walks

  • regularity method

approximation by an (n × n)-matrix A rows and columns ≈ parts in regularity decomposition Aij ≥ 0 and Aij + Aji = 1 for all i, j ∈ {1, . . . , n}

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Cases of two and three parts

  • non-negative matrix A, s.t. A + AT = J
  • properties of the spectrum of A:

Tr A = λ1 + . . . + λk = 1/2 Perron–Frobenius ⇒ ∃ρ ∈ R : ρ = λ1 and |λi| ≤ λ1 v∗(A + AT )v = v∗(λi + λi)v = v∗Jv ≥ 0 ⇒ Re λi ≥ 0

  • fix Tr A3 = λ3

1 + . . . + λ3 k ∈ [1/36, 1/8]

minimize Tr A4 = λ4

1 + . . . + λ4 k

  • optimum λ≤k−1 = ρ and λk = 1/2 − (k − 1)ρ, k ∈ {2, 3}

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Case of two parts—structure

  • A = (J + B)/2, B is antisymmetric, i.e. B = −BT

A is non-negative and A + AT = J

  • analysis of antisymmetric matrix B

σi and αi for matrix B with

i cos2 αi = 1

B = U T        σ1 −σ1 σ2 −σ2        U

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Case of two parts—structure

  • A = (J + B)/2, B is antisymmetric, i.e. B = −BT

A is non-negative and A + AT = J

  • analysis of antisymmetric matrix B

σi and αi for matrix B with

i cos2 αi = 1

  • Tr A3 ≈ Tr J3 + Tr JB2 =

i σ2 i cos2 αi

Tr A4 ≈ Tr J4 + Tr J2B2 + Tr B4 ≈ Tr JB2 +

i σ4 i

  • optimum for α1 = 0, α≥2 = π/2 and σ≥2 = 0

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Case of two parts—structure

  • A = (J + B)/2, B is antisymmetric, i.e. B = −BT

σi and αi for matrix B with

i cos2 αi = 1

  • ptimum for α1 = 0, α≥2 = π/2 and σ≥2 = 0
  • assign pv ∈ [0, 1/2] to each vertex v
  • rient from v to w with probability 1/2 + (pv − pw)
  • conjectured construction: pv ∈ {0, 1/2}

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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

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Thank you for your attention!

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