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# Tournaments with near-linear transitive subsets

## Author(s): Choromanski, Krzysztof; Chudnovsky, Maria; Seymour, Paul D.

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 Abstract: Let H be a tournament, and let epsilon >= 0 be a real number. We call an “Erdos-Hajnal coefficient” for H if there exists c > 0 such that in every tournament G not containing H as a subtournament, there is a transitive subset of cardinality at least c vertical bar V(G)vertical bar(epsilon). The Erdos-Hajnal conjecture asserts, in one form, that every tournament H has a positive Erdos-Hajnal coefficient. This remains open, but recently the tournaments with Erdos-Hajnal coefficient 1 were completely characterized. In this paper we provide an analogous theorem for tournaments that have an Erdos-Hajnal coefficient larger than 5/6; we give a construction for them all, and we prove that for any such tournament H there are numbers c, d such that, if a tournament G with vertical bar V(G)vertical bar > 1 does not contain H as a subtournament, then V(G) can be partitioned into at most c(log(vertical bar V(G)vertical bar))(d) transitive subsets. (C) 2014 Elsevier Inc. All rights reserved. Publication Date: Nov-2014 Electronic Publication Date: 8-Jul-2014 Citation: Choromanski, Krzysztof, Chudnovsky, Maria, Seymour, Paul. (2014). Tournaments with near-linear transitive subsets. JOURNAL OF COMBINATORIAL THEORY SERIES B, 109 (228 - 249. doi:10.1016/j.jctb.2014.06.007 DOI: doi:10.1016/j.jctb.2014.06.007 ISSN: 0095-8956 EISSN: 1096-0902 Pages: 228 - 249 Type of Material: Journal Article Journal/Proceeding Title: JOURNAL OF COMBINATORIAL THEORY SERIES B Version: Author's manuscript

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