Heath 2026 generalized ramsey numbers hypercube

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E. Heath, C. Schwieder and S. Zerbib, Generalized Ramsey numbers in the hypercube, arXiv:2601.15451v1 [math.CO] 21 January 2026, 12 pages; a preprint (the arXiv API record of 2026-09-18 lists one version and no journal reference).

The retained folder-name PDF is that v1 (12 pages, with a text layer). Pages 1--2 were read on the page images. The arXiv record (https://arxiv.org/abs/2601.15451, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status: claims checked for the abstract's main result and for Theorems 2 and 3 (p. 2), read clause by clause on the page images; the proofs were not read.

The paper studies the generalized Ramsey number $f(Q_n,C_k,q)$, the least number of colors in an edge-coloring of the hypercube $Q_n$ in which every copy of the cycle $C_k$ receives at least $q$ colors; this is the Erdős--Shelah function $f(G,H,q)$ with host $G=Q_n$ and $H=C_k$, not the Ramsey number $r(Q_n)$ of Problem 181. The introduction (pp. 1--2) recalls the Erdős--Gyárfás bound $f(K_n,K_p,q)=O(n^{(p-2)/(\binom p2-q+1)})$, the Bennett--Delcourt--Li--Postle improvement (1), Faudree, Gyárfás, Lesniak and Schelp's $f(Q_n,C_4,4)=n$ for $n=4$ or $n\ge6$ (display (2)), the Mubayi--Stading bounds $c_1n^{k/4}\le f(Q_n,C_k,k)\le c_2n^{k/4}$ for $4\mid k$ and $3n-2\le f(Q_n,C_6,6)\le n^{1+o(1)}$ (their Theorem 1), and Conder's $f(Q_n,C_6,2)\le3$. Theorem 2 (p. 2): for integers $k\ge3$ and $3\le q\le k+1$, $f(Q_n,C_{2k},q)=o\bigl(n^{(k-1)/(2k-q+1)}\bigr)$, proved by the bipartite conflict-free matching method (the abstract states the same with $k$ for $2k$: $k\ge6$, $3\le q\le k/2+1$, $f(Q_n,C_k,q)=o(n^{(k/2-1)/(k-q+1)})$). Theorem 3 (p. 2): $f(Q_n,C_6,4)>(n-1)^{1/3}$ and $f(Q_n,C_6,5)>(n-1)^{1/2}$. The paper bounds a different quantity from $r(Q_n)$ and contains no statement about the Ramsey number of the hypercube; it was consulted for Problem 181 as a 2026 paper on hypercube colorings, as context only.

Contents

• Abstract and introduction (pp. 1--2): the definition of $f(G,H,q)$; the Erdős--Gyárfás and Bennett--Delcourt--Li--Postle bounds; display (2), Theorem 1 (Mubayi--Stading) and Conder's $3$-coloring, all quoted from the literature.
• Theorem 2 (p. 2): $f(Q_n,C_{2k},q)=o\bigl(n^{(k-1)/(2k-q+1)}\bigr)$ for $k\ge3$ and $3\le q\le k+1$.
• Theorem 3 (p. 2): $f(Q_n,C_6,4)>(n-1)^{1/3}$ and $f(Q_n,C_6,5)>(n-1)^{1/2}$.
• Later sections (pp. 3--12; not read): further bounds for $4$-cycles and $6$-cycles and the proofs.

Compiled scope

Pages 1--2 were read on the page images; pp. 3--12 were not read. No proof was checked and nothing here is independently reviewed.

Source: https://arxiv.org/abs/2601.15451.

Bears on. #181: context only. The quantity $f(Q_n,C_k,q)$ is a coloring number of the hypercube's edges, not its Ramsey number; nothing in the paper bounds $r(Q_n)$, and it is recorded on the problem page as evidence of continued work on hypercube colorings, not as progress.