The problem
For every k and δ > 0 there exists N such that any subset of [k]^N of size ≥ δ·\(k^{N}\) contains a combinatorial line (a set of k points varying together across coordinates).
The Density Hales–Jewett Theorem (Polymath One)
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History & significance
Furstenberg and Katznelson proved DHJ via ergodic theory in 1991 — existence without usable bounds.
The resolution (human proof)
Solved by Polymath One (2009-2010), published 2012. Tim Gowers proposed attacking the k=3 case openly on his blog in January 2009, birthing the Polymath experiment: over two dozen contributors and a thousand comments later DHJ(3) fell, with the first effectively-bounded proof and the general statement completed alongside (the collective paper 'Density Hales-Jewett and Moser numbers' appeared in 2012). The experiment itself proved durable — Polymath projects later cracked bounded prime gaps, the Erdős discrepancy problem and more, several already catalogued here.