Erdős Problem #75

Is there a graph of chromatic number $\aleph_1$ such that for all $\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\epsilon}$?

#75: [EHS82][Er95][Er95d]

graph theory | chromatic number

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Conjectured by Erdős, Hajnal, and Szemerédi [EHS82]. In [Er95d] Erdős suggests this may even be true with an independent set of size $\gg n$.

See also [750].

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