Erdős Problem #75

The problem

Is there a graph of chromatic number $\aleph_1$ with $\aleph_1$ vertices 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 $>n^{1-\epsilon}$?

What about an independent set of size $\gg n$?

Statement quoted from erdosproblems.com/75, maintained by Thomas Bloom. Original source: [EHS82,p.120][Er95,p.11][Er95d,p.63]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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