TY - JOUR

T1 - Pure Pairs. II. Excluding All Subdivisions of A Graph

AU - Chudnovsky, Maria

AU - Scott, Alex

AU - Seymour, Paul

AU - Spirkl, Sophie

N1 - Funding Information:
Supported by a Leverhulme Trust Research Fellowship.
Funding Information:
Supported by ONR grant N00014-14-1-0084, AFOSR grant A9550-19-1-0187, and NSF grants DMS-1265563 and DMS-1800053.
Funding 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.
Publisher Copyright:
© 2021, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

PY - 2021/6

Y1 - 2021/6

N2 - 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, B ⊆ V(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.

AB - 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, B ⊆ V(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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U2 - 10.1007/s00493-020-4024-1

DO - 10.1007/s00493-020-4024-1

M3 - Article

AN - SCOPUS:85109637063

SN - 0209-9683

VL - 41

SP - 379

EP - 405

JO - Combinatorica

JF - Combinatorica

IS - 3

ER -