On R\"odl's Theorem for Cographs
Lior Gishboliner, Asaf Shapira

TL;DR
This paper provides a concise proof of a generalized version of R"odl's theorem for cographs, improving understanding of induced $F$-free graphs and their vertex subsets with specific edge densities.
Contribution
It offers a shorter proof of a broader statement related to R"odl's theorem, extending results to cographs and graphs with few $P_4$ copies.
Findings
Established a more general version of R"odl's theorem for cographs.
Provided a shorter proof technique for the theorem.
Extended the theorem's applicability to graphs with few $P_4$ copies.
Abstract
A theorem of R\"odl states that for every fixed and there is so that every induced -free graph contains a vertex set of size whose edge density is either at most or at least . R\"odl's proof relied on the regularity lemma, hence it supplied only a tower-type bound for . Fox and Sudakov conjectured that can be made polynomial in , and a recent result of Fox, Nguyen, Scott and Seymour shows that this conjecture holds when . In fact, they show that the same conclusion holds even if contains few copies of . In this note we give a short proof of a more general statement.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications · Advanced Graph Theory Research
