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Claim. Let $f$ be a non-constant polynomial with integer coefficients that takes an even value at some integer. Then for every finite coloring of the integers the equation $x+y=f(z)$ has a solution with $x$ and $y$ of the same color and $x\ne y$. The squares are the case $f(z)=z^2$ and the $k$th powers the case $f(z)=z^k$, which takes the even value $2^k$, so the theorem answers both parts of the question affirmatively. The distinctness of $x$ and $y$ is what gives the theorem content: the even value $f(z_0)$ always has the equal-summand solution $x=y=f(z_0)/2$. The word non-constant is needed with it: a constant $f\equiv2m$ has colorings with no two distinct integers of one color summing to $2m$ (color $n$ by the sign of $n-m$). The publisher's abstract states the theorem without the clause $x\ne y$ and without the word non-constant, which the site's commentary supplies; Sanders's refereed 2020 note restates it for the squares with distinct $x$, $y$ in the finite form on $\{1,\ldots,N\}$, which implies the infinite one, and for a general $f$, including $z^k$, the clause rests on the site's account and on the paper's title, which announces a count of monochromatic solutions from which nontrivial ones follow. The partial result before it, Theorem 3 of Erdős, Sárközy and Sós (1989) for at most three colors, has its own page, Erdős, Sárközy and Sós 1989.

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

Acceptance. Refereed: A. Khalfalah and E. Szemerédi, On the number of monochromatic solutions of $x+y=z^2$, Combin. Probab. Comput. 15 (2006), no. 1--2, 213--227, published online 3 January 2006 (Crossref record), the date this page is named by. Later refereed papers cite it: Sanders (Acta Math. Hungar. 161 (2020)) as the answer to a question of Roth, Erdős, Sárközy and Sós, and Green and Lindqvist (Canad. J. Math. 71 (2019)) in a remark (p. 580) that states it without the condition $x\ne y$, filed as Sanders 2020 and Green and Lindqvist 2019. Reviewed: the site's curator, T. F. Bloom, labels the problem PROVED and credits the theorem, in the problem's commentary, in the general form with a non-constant $f$ (page last edited 7 April 2026); the thread and proof-claim tab are empty. None of the 23 citing papers that Semantic Scholar listed on 2026-09-18 disputes the theorem.

Sources of the statement. The paper is closed access and not held, so its theorem is cited here through the publisher's abstract, the introduction of Sanders's note and the remark of Green and Lindqvist; the $k$th-power clause rests on the abstract's general $f$ and the site's commentary.