发表机构
Center for Discrete Mathematics, Fuzhou University; Center of pure Mathematics, School of Mathematical Sciences, Anhui University（福州大学离散数学中心; 安徽大学数学科学学院纯数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了弱d-退化的有序3-图H的有序Ramsey数上界为$t\\,2^{C_d n^{2-c/d}}$，解决了Balko和Vizer的问题，并表明弱退化性假设不能替换为标准退化性。
AI 中文摘要
有序k-图G和H的有序Ramsey数$r_<(G,H)$是使得在$[N]$上自然有序的完全k-图的每个红蓝边着色包含一个蓝色有序副本G或一个红色有序副本H的最小整数N。我们证明存在一个绝对常数$c>0$，使得对于每个整数$d\ge1$，存在一个常数$C_d>0$，使得每个在t个顶点上的弱d-退化的有序3-图H满足对于每个正整数n，$r_<\bigl(H,K_3^{(3)}(n)\bigr) \le t\\,2^{C_d n^{2-c/d}}$。这以更强的形式解决了Balko和Vizer（{\em SIAM J. Discrete Math., 2022}）提出的一个问题。此外，我们证明弱退化性假设不能被有界标准退化性所取代。特别地，对于每个足够大的n，存在一个在至多$2^{O(n)}$个顶点上的1-退化的有序3-图F，使得$r_<\bigl(F,K_3^{(3)}(n)\bigr)>2^{\Omega(n^2)}$。
英文摘要
The \emph{ordered Ramsey number} $r_<(G,H)$ of ordered $k$-graphs $G$ and $H$ is the least integer $N$ such that every red-blue edge-coloring of the naturally ordered complete $k$-graph on $[N]$ contains a blue ordered copy of $G$ or a red ordered copy of $H$. We prove that there is an absolute constant $c>0$ such that, for every integer $d\ge1$, there is a constant $C_d>0$ for which every weakly $d$-degenerate ordered $3$-graph $H$ on $t$ vertices satisfies \[ r_<\bigl(H,K_3^{(3)}(n)\bigr) \le t\,2^{C_d n^{2-c/d}} \] for every positive integer $n$. This resolves a problem posed by Balko and Vizer ({\em SIAM J. Discrete Math., 2022}) in a stronger form. Furthermore, we show that the weak-degeneracy hypothesis cannot be replaced by bounded standard degeneracy. In particular, for every sufficiently large $n$, there exists a $1$-degenerate ordered $3$-graph $F$ on at most $2^{O(n)}$ vertices such that $r_<\bigl(F,K_3^{(3)}(n)\bigr)>2^{Ω(n^2)}.$
