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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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Erdős–Hajnal for graphs with no 5‐hole
Abstract The Erdős–Hajnal conjecture says that for every graph there exists such that every graph not containing as an induced subgraph has a clique or stable set of cardinality at least . We prove that this is true when is a cycle of length five. We also prove several further results: for instance, that if is a cycle and is the complement of a forest, there exists such that every graph containing neither of as an induced subgraph has a clique or stable set of cardinality at least .
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- PAR ID:
- 10443973
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 126
- Issue:
- 3
- ISSN:
- 0024-6115
- Page Range / eLocation ID:
- p. 997-1014
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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