PROVED
This has been solved in the affirmative.
Is it true that for every $\epsilon>0$ and integer $t\geq 1$, if $N$ is sufficiently large and $A$ is a subset of $[t]^N$ of size at least $\epsilon t^N$ then $A$ must contain a combinatorial line $P$ (a set $P=\{p_1,\ldots,p_t\}$ where for each coordinate $1\leq j\leq t$ the $j$th coordinate of $p_i$ is either $i$ or constant).
The 'density Hales-Jewett' problem. This was proved by Furstenberg and Katznelson
[FuKa91]. A new elementary proof, which gives quantitative bounds, was proved by the Polymath project
[Po09].
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T. F. Bloom, Erdős Problem #171, https://www.erdosproblems.com/171, accessed 2026-01-16
