Tournaments and the strong Erdős–Hajnal Property

Eli Berger; Krzysztof Choromanski; Maria Chudnovsky; Shira Zerbib

doi:10.1016/j.ejc.2021.103440

Tournaments and the strong Erdős–Hajnal Property

Eli Berger, Krzysztof Choromanski, Maria Chudnovsky, Shira Zerbib

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős–Hajnal Conjecture, states that for every tournament S there exists ɛ(S)>0 such that if T is an n-vertex tournament that does not contain S as a subtournament, then T contains a transitive subtournament on at least n^ɛ(S) vertices. Let C₅ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon–Pach–Solymosi conjecture is known to be true for the case when S=C₅. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C₅ there exist disjoint vertex subsets A and B, each containing a linear proportion of the vertices of T, and such that every vertex of A is adjacent to every vertex of B.

Original language	English
Article number	103440
Journal	European Journal of Combinatorics
Volume	100
DOIs	https://doi.org/10.1016/j.ejc.2021.103440
State	Published - Feb 2022

Bibliographical note

Funding Information:
Eli Berger, Maria Chudnovsky and Shira Zerbib were supported by US-Israel BSF grant, Israel2016077.Maria Chudnovsky was supported by National Science Foundation, United States of America grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant Number W911NF1610404.Shira Zerbib was supported by National Science Foundation, United States of America grant DMS-1953929.

Publisher Copyright:
© 2021 Elsevier Ltd

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1016/j.ejc.2021.103440

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title = "Tournaments and the strong Erd{\H o}s–Hajnal Property",

abstract = "A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd{\H o}s–Hajnal Conjecture, states that for every tournament S there exists ɛ(S)>0 such that if T is an n-vertex tournament that does not contain S as a subtournament, then T contains a transitive subtournament on at least nɛ(S) vertices. Let C5 be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon–Pach–Solymosi conjecture is known to be true for the case when S=C5. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C5 there exist disjoint vertex subsets A and B, each containing a linear proportion of the vertices of T, and such that every vertex of A is adjacent to every vertex of B.",

author = "Eli Berger and Krzysztof Choromanski and Maria Chudnovsky and Shira Zerbib",

note = "Funding Information: Eli Berger, Maria Chudnovsky and Shira Zerbib were supported by US-Israel BSF grant, Israel2016077.Maria Chudnovsky was supported by National Science Foundation, United States of America grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under Grant Number W911NF1610404.Shira Zerbib was supported by National Science Foundation, United States of America grant DMS-1953929. Publisher Copyright: {\textcopyright} 2021 Elsevier Ltd",

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N2 - A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős–Hajnal Conjecture, states that for every tournament S there exists ɛ(S)>0 such that if T is an n-vertex tournament that does not contain S as a subtournament, then T contains a transitive subtournament on at least nɛ(S) vertices. Let C5 be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon–Pach–Solymosi conjecture is known to be true for the case when S=C5. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C5 there exist disjoint vertex subsets A and B, each containing a linear proportion of the vertices of T, and such that every vertex of A is adjacent to every vertex of B.

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Tournaments and the strong Erdős–Hajnal Property

Abstract

Bibliographical note

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