Erdős Problem #922

Let $k\geq 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of size $\geq (n-k)/2$, where $n$ is the number of vertices of $H$. Must $G$ have chromatic number at most $k+2$?

#922: [ErHa67b][Er69b]

graph theory | chromatic number

A question of Erdős and Hajnal [ErHa67b]. The case $k=0$ is trivial, but they could not prove this even for $k=1$.

This is true, and was proved by Folkman [Fo70b].

See also [73].

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Additional thanks to: Raphael Steiner

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