Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs

@article{Chudnovsky2020PurePI,
  title={Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs},
  author={M. Chudnovsky and Jacob Fox and Alex D. Scott and Paul D. Seymour and Sophie Theresa Spirkl},
  journal={J. Graph Theory},
  year={2020},
  volume={95},
  pages={315-340}
}
A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has degree at least $sn$, or there are two disjoint sets of vertices, of sizes at least $sn^s$ and $sn$, anticomplete to each other. We prove this holds for a large class of graphs $H$, and we prove that something like it holds for all graphs $H$. Say $H$ is… Expand
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