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.
Bibliographical noteFunding 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.
© 2021 Elsevier Ltd
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics