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Claim. For every positive integer $n$, the vertex set of every red--blue edge-colored complete graph on $n$ vertices can be covered by at most $\sqrt n$ monochromatic paths, all of the same color; this is Theorem 1.5 of Chen and Chen (p. 2 of arXiv:2607.21915v1), the statement of Problem 518 with no restriction on $n$. The preprint argues by a minimal counterexample (Section 3, pp. 3--13), an argument that runs over every $n$ and does not use the large-$n$ theorem; the summary filed with it on the site describes it as closing the small values of $n$ that the large-$n$ theorem leaves open, so that no restriction on $n$ remains. Section 2 (pp. 2--3) states without proof the three lemmas it takes from Pokrovskiy, Versteegen and Williams (the preprint's Lemmas 2.3--2.5, which are Lemmas 2.2, 2.4 and 3.3 of that paper: two lemmas on paths in bipartite graphs, and a bound on the size of a blue path cover of $K_n$ built on a given blue path), the cut-coloring lemma of Erdős and Gyárfás (Lemma 2.1) and the monochromatic path on $\lfloor2n/3\rfloor+1$ vertices of Gerencsér and Gyárfás (Lemma 2.2); Section 3 applies the three borrowed lemmas six times.

Submission note. Posted to erdosproblems.com as a proof claim by Hangdi Chen, Yaojun Chen (account NicoV) on 29 July 2026:

This paper rules out counterexamples for all small remaining values of n, thereby proving that the result holds exactly for every n≥1.

Standing. Claimed. The preprint is arXiv v1 of 24 July 2026, the only version on 2026-09-18, with no journal reference on the arXiv record. A site user submitted it to the site's proof-claims tab on 2026-07-29 as a full proof by the authors; the entry has no comments, and the site's label and commentary do not mention it: the site's PROVED attaches to the large-$n$ theorem of Pokrovskiy, Versteegen and Williams 2024, which the preprint's introduction cites as its Theorem 1.4 and never uses; what the proof takes from that paper is the three lemmas named above, so the argument is not self-contained. No citing paper was found in the search the problem page records, and no check of the proof is recorded. A refereed version or a documented independent acceptance would move the claim to accepted, and the problem's standing with it.

Depends on. Gerencsér and Gyárfás, Theorem 1, whose diagonal case is the preprint's Lemma 2.2. The minimal-counterexample argument runs over every $n$ and does not rest on the large-$n$ theorem recorded on the claim page of Pokrovskiy, Versteegen and Williams; the three lemmas it takes from their paper without proof, and the cut-coloring lemma of Erdős and Gyárfás, have no claim page, theory card or result page here.