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Claim. The answer to Problem 658 is yes, as a consequence of the density Hales–Jewett theorem of H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem. In the form Solymosi records as his Theorem 3.1, the theorem gives, for every $\delta>0$ and positive integers $K$ and $d$, an $N_0(\delta,K,d)$ such that for $N>N_0$ every subset of $[N]^d$ of size at least $\delta N^d$ contains a homothetic copy of $[K]^d$; with $d=K=2$ the copy $\{(a,b),(a+t,b),(a,b+t),(a+t,b+t)\}$, $t\ge1$, is an axis-parallel square, which is the site's question in Graham's form and so in the form allowing any square. The proof is ergodic and gives no bound on $N_0$. The library holds no copy of the paper; the statement is given in the form Solymosi's paper records, as on the Solymosi card, and as the site's commentary credits it.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: J. Analyse Math. 57 (1991), 64--119, doi:10.1007/BF03041066; the Crossref record dates the volume to December 1991, filled to the first of the month for this page's name. Reviewed: the site's curator, Thomas Bloom, labels the problem proved and credits the answer without a bound to Furstenberg and Katznelson's density Hales–Jewett theorem in the problem page's commentary; the proof-claim tab is empty. Solymosi's later quantitative proof gives the same conclusion with a bound that is at best of tower type.