TY - JOUR

T1 - Pure pairs. X. Tournaments and the strong Erdős-Hajnal property

AU - Chudnovsky, Maria

AU - Scott, Alex

AU - Seymour, Paul

AU - Spirkl, Sophie

N1 - Publisher Copyright:
© 2023 Elsevier Ltd

PY - 2024/1

Y1 - 2024/1

N2 - A pure pair in a tournament G is an ordered pair (A,B) of disjoint subsets of V(G) such that every vertex in B is adjacent from every vertex in A. Which tournaments H have the property that if G is a tournament not containing H as a subtournament, and |G|>1, there is a pure pair (A,B) in G with |A|,|B|≥c|G|, where c>0 is a constant independent of G? Let us say that such a tournament H has the strong EH-property. As far as we know, it might be that a tournament H has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdős-Hajnal conjecture, which in one form says that for every tournament H there exists τ>0 such that every tournament G not containing H as a subtournament has a transitive subtournament of cardinality at least |G|τ. Let us say that a tournament H satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.

AB - A pure pair in a tournament G is an ordered pair (A,B) of disjoint subsets of V(G) such that every vertex in B is adjacent from every vertex in A. Which tournaments H have the property that if G is a tournament not containing H as a subtournament, and |G|>1, there is a pure pair (A,B) in G with |A|,|B|≥c|G|, where c>0 is a constant independent of G? Let us say that such a tournament H has the strong EH-property. As far as we know, it might be that a tournament H has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdős-Hajnal conjecture, which in one form says that for every tournament H there exists τ>0 such that every tournament G not containing H as a subtournament has a transitive subtournament of cardinality at least |G|τ. Let us say that a tournament H satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.

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U2 - 10.1016/j.ejc.2023.103786

DO - 10.1016/j.ejc.2023.103786

M3 - Article

AN - SCOPUS:85166772299

SN - 0195-6698

VL - 115

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103786

ER -