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Claim. N. Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227--245. Theorem 4.4 (p. 244): every partition of $\{2,\ldots,990\}$ into two cells has distinct $x,y$ with $\{x,y,x+y,xy\}$ in one cell, and $990$ cannot be lowered. Theorem 4.3 (p. 243) is credited to Graham's unpublished computation, which Hindman verified independently; it gives the same for $\{1,\ldots,252\}$, with $252$ sharp. Either theorem gives every $2$-coloring of $\mathbb N$ a set $A=\{x,y\}$ whose sums and products of distinct elements share one color: $x$, $y$, $x+y$ and $xy$, with the singletons included as the problem page reads them. Both theorems rest on computer searches: Theorem 4.3 lists its 27 cases and checks one in print, and Theorem 4.4 prints only the extremal partition of $\{2,\ldots,989\}$. Erdős reported both computations in 1977 (p. 58). Bowen's Theorem 1.1 at $n=2$ (Adv. Math. 462 (2025), 110095; arXiv:2205.12921, 25 May 2022) reproves the case without a computer and gives arbitrarily large such $x,y$. It settles the same instance and has no page of its own.

Covers. The case of two colors and $|A|=2$. Not covered: three or more colors, and $|A|\ge3$.

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

Acceptance. Refereed: Trans. Amer. Math. Soc. 247 (1979), 227--245. The computer searches were not rerun here. The site labels the problem OPEN and cites neither this paper nor Bowen's, so no curator credit is listed.