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Pure Pairs. II. Excluding All Subdivisions of A Graph

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Abstract

We prove for every graph H there exists ɛ > 0 such that, for every graph G with |G|≥2, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least ɛ|G| neighbours, or there are two disjoint sets A, BV(G) with |A|,|B|≥ɛ|G| such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|c. This is related to the Erdős-Hajnal conjecture.

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Correspondence to Alex Scott.

Additional information

Supported by NSF grant DMS-1550991. 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 W911NF-16-1-0404.

Supported by a Leverhulme Trust Research Fellowship.

Supported by ONR grant N00014-14-1-0084, AFOSR grant A9550-19-1-0187, and NSF grants DMS-1265563 and DMS-1800053.

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Chudnovsky, M., Scott, A., Seymour, P. et al. Pure Pairs. II. Excluding All Subdivisions of A Graph. Combinatorica 41, 379–405 (2021). https://doi.org/10.1007/s00493-020-4024-1

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  • DOI: https://doi.org/10.1007/s00493-020-4024-1

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