The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundstr\"om~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell}+1, \] and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor $k^{k/\ell}$ to $(k!)^{1/\ell}$. As Jenssen and Skokan (AM, 2021) determined $R_k(C_{2\ell+1})$ for fixed $k$ and sufficiently large $\ell$, it becomes even more interesting to seek better bound for fixed $\ell$ and sufficiently large $k$. In this paper, we show \[ R_k(C_{2\ell+1}) \le \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell}+O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1 \] for every fixed $\ell\ge 2$ and sufficiently large $k$, which improves the bound of Miyazaki et al. by a factor $2^{k-o(k)}$, and the bound of Axenovich et al. by a factor $(2\e^{1/\ell})^{k-o(k)}$.
In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycle...
For $k\geq2$, let $S_r(k;\ell)$ be the smallest $n$, if exists, such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution $\mathcal{S}$ to the equation \[ x_1+x_2+\cdots+x_k=x_{k+1} \] such that the number of distinct integers in $\mathcal{S}$ is exactly $\ell+1$. We prove that, if $\ell\geq2$ is...
This work generalizes and combines tools from the $(k+2)-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023] and yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.
The size-Ramsey number $\widehat{R}_k(\mathcal G)$ of a $k$-uniform hypergraph $\mathcal G$ is the minimum number of edges in a $k$-uniform hypergraph $\mathcal H$ such that every $2$-edge-coloring of $\mathcal H$ contains a monochromatic copy of $\mathcal G$. The following question was pointed out by Fox and recorded...
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$....
Given a coloring $c$ and an even $k\ge 4$, a nontrivial $k$-term arithmetic progression~($k$-AP) $a,a+d,\ldots,a+(k-1)d$ is called symmetrically colored if $c(a+(i-1)d)=c(a+(k-i)d)$, $\forall i\in[k/2]$. Deng, Tidor, and Zhao asked whether $[N]$ admits a coloring with $N^{o(1)}$ colors and no such 4-APs, and gave an $O...
Ruizhe Shi, Yiqi Dong· arXiv.org· 0 citations
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