发表机构
Tianjin Normal University（天津师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对固定2≤ℓ<k的大小拉姆齐数问题，研究人员在ℓ≥3且ℓ+1≤k≤2ℓ-2的范围内构造了特定k-均匀松弛ℓ-树，证明其大小拉姆齐数非多项式，否定了该范围的多项式有界性猜想。
AI 中文摘要
k-均匀超图𝒢的大小拉姆齐数$\u0302R_k(𝒢)$是指满足以下条件的k-均匀超图ℋ的最小边数：ℋ的每一种2-边着色都包含𝒢的单色副本。Fox提出了一个问题，该问题由Dudek、La Fleur、Mubayi和Rödl记录：对于固定的2≤ℓ<k，每个k-均匀松弛ℓ-树的大小拉姆齐数是否被n的多项式所界定？我们在ℓ≥3且ℓ+1≤k≤2ℓ-2的范围内回答了这个问题。对于每个足够大的n，我们构造了一个恰有n个顶点的k-均匀松弛ℓ-树$\u0305𝒯_{n,ℓ}^{(k)}$，使得$\u0302R_k(\u0305𝒯_{n,ℓ}^{(k)})≥2^{c_{k,ℓ}n^{1/ℓ}}$，其中$c_{k,ℓ}>0$是仅依赖于k和ℓ的常数。
英文摘要
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 by Dudek, La Fleur, Mubayi and Rödl~\cite{Dudek-Fleur-Mubayi-Rodl}: for fixed $2\le \ell<k$, is the size-Ramsey number of every $k$-uniform relaxed $\ell$-tree bounded by a polynomial in $n$? We answer this question in the range \[ \ell\geq3 \quad\text{and}\quad \ell+1\leq k\leq2\ell-2. \] For every sufficiently large $n$, we construct a $k$-uniform relaxed $\ell$-tree $\bar{\mathcal{T}}_{n,\ell}^{(k)}$ on exactly $n$ vertices such that $$ \widehat{R}_k(\bar{\mathcal{T}}_{n,\ell}^{(k)})\ge 2^{c_{k,\ell}n^{1/\ell}} $$ for a constant $c_{k,\ell}>0$ depending only on $k$ and $\ell$.
Comments7 pages
