Year: 2026
Type: preprint
Abstract: 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 $\mathcal G$. Fox asked whether, for fixed $2\le \ell0$ depends only on $k$ and $\ell$. Moreover, any two distinct edges of the construction intersect in at most $2\ell-k-1$ vertices. In contrast, for $k\ge 2\ell$, we prove that every $k$-uniform relaxed $\ell$-tree $\bar{\mathcal T}_{n,\ell}^{(k)}$ on $n$ vertices satisfies \[ \widehat{R}_k\bigl(\bar{\mathcal T}_{n,\ell}^{(k)}\bigr) =O_{k,\ell}(n^{2k}). \]
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211900258