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Abstract In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ has a clique or stable set of size at least $$|G|^{c}$$, and they proved that this is true with $$ |G|^{c}$$ replaced by $$2^{c\sqrt{\log |G|}}$$. Until now, there has been no improvement on this result (for general $$H$$). We prove a strengthening: that for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ with $$|G|\ge 2$$ has a clique or stable set of size at least $$ \begin{align*} &2^{c\sqrt{\log |G|\log\log|G|}}.\end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.
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This content will become publicly available on May 1, 2026
Even pairs in Berge graphs with no balanced skew-partitions
Let G be a Berge graph that has no odd prism and no antihole of length at least six as an induced subgraph. We show that every such graph G with no balanced skew-partition is either complete or has an even pair.
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- Award ID(s):
- 2120644
- PAR ID:
- 10613183
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Discrete Mathematics
- Volume:
- 348
- Issue:
- 5
- ISSN:
- 0012-365X
- Page Range / eLocation ID:
- 114388
- Subject(s) / Keyword(s):
- Discrete Mathematics Berge graphs Coloring Even pairs Induced subgraphs
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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