AI 中文总结
本文解决了零和拉姆齐数相关的两个猜想和一个问题，确定了偶数边$\delta$集的$\mathbb{Z}_2$零和拉姆齐数，证明双树不满足Caro的一个猜想，并构造无限族树肯定回答了Caro关于$R(T,\mathbb{Z}_m)>R(T,2)$的问题。
AI 中文摘要
设$K_N^{(r)}$表示顶点数为$N$的完全$r$均匀超图。对于一个$r$均匀超图$H$和整数$k\geq2$，$k$色拉姆齐数$R(H,k)$是满足以下条件的最小整数$N$：对$K_N^{(r)}$的任意$k$边着色，都包含一个单色的$H$副本。当$k$整除$H$的边数$\vert E(H)\vert$时，零和拉姆齐数$R(H,\mathbb{Z}_k)$是满足以下条件的最小整数$N$：对$K_N^{(r)}$的每条边赋予$\mathbb{Z}_k$中的一个元素作为标签，都包含一个$H$副本，其边标签在$\mathbb{Z}_k$中的和为$0$。本文解决了与这两种拉姆齐数相关的两个猜想和一个问题。首先，Caro和Provstgaard提出了边数为偶数的$\delta$集在$\mathbb{Z}_2$上的零和拉姆齐数的精确值，本文确定了这些数值，从而证明了他们的猜想。其次，对于边数为$m$的森林$F$，令$tF$表示$t$个$F$的不交并，Caro猜想当$t$足够大时，$R(tF,\mathbb{Z}_{mt})=R(tF,2)$，本文证明该猜想对双树不成立。Caro还询问是否存在边数为$m$的树$T$，使得$R(T,\mathbb{Z}_m)>R(T,2)$，本文通过构造此类树的无限族，肯定回答了该问题。
英文摘要
Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\geq2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramsey number $R(H,\mathbb Z_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of $\mathbb Z_k$ contains a copy of $H$ whose edge labels sum to $0$ in $\mathbb Z_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over $\mathbb Z_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,\mathbb Z_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,\mathbb Z_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.
