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Claim. For any partition of the positive integers into at most three classes there are infinitely many squares $x^2=a_1+a_2$ with $a_1\ne a_2$ in one class. This is Theorem 3 of P. Erdős, A. Sárközy and V. T. Sós, On a conjecture of Roth and some related problems. I, in Irregularities of Partitions, Algorithms and Combinatorics 8, Springer (1989), 47--59, printed p. 55 (result page Theorem 3; source card Erdős, Sárközy and Sós 1989). The proof takes, by the paper's Lemma 2, an integer with three representations as a sum of two nearly equal squares, solves the six equations $u_i+u_j=$ (the six squares) in four distinct positive numbers $u_1,\ldots,u_4$, and observes that with at most three classes two of the four share a class, so one of the six squares is a monochromatic sum of distinct summands. The authors introduce the theorem by saying that their result is not strong enough to give, for an arbitrary number of classes, a monochromatic solution of $a_1+a_2=x^2$ with $a_1\ne a_2$ (p. 54).

Covers. The square question of Problem 439 for colorings with two or three colors: every such coloring has two distinct integers of one color whose sum is a square, and infinitely many squares arise this way. The general case of the square question and the $k$th-power question are not covered; both were settled later by Khalfalah and Szemerédi 2006.

Depends on. Nothing in this wiki; the result rests on the cited chapter alone.

Acceptance. None listed. The chapter is part of a conference volume whose refereeing is not documented in its Crossref record or on the library card, so refereed is not listed. The site's curator, T. F. Bloom, labels the problem PROVED and credits that label to Khalfalah and Szemerédi; the commentary's sentence that Erdős, Sárközy and Sós proved the statement for two or three colors (page last edited 7 April 2026) mentions this result but is not the label's credit, so reviewed is not listed either. The volume gives no day of publication, so the day in the page name is a placeholder for 1989.