Anti-Ramsey number of matchings in $3$-uniform hypergraphs

Abstract

Let $n,s,$ and $k$ be positive integers such that $k\ge 3$, $s\ge 3$ and $n\ge ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\text{ar}(n,k,M_s)$ of an $s$-matching is the smallest integer $c$ such that each edge-coloring of the $n$-vertex $k$-uniform complete hypergraph with exactly $c$ colors contains an $s$-matching with distinct colors. In 2013, Özkahya and Young proposed a conjecture on the exact value of ar$(n,k,M_s)$ for all $n\ge sk$ and $k\ge 3$. A 2019 result by Frankl and Kupavskii verified this conjecture for all $n\ge sk+(s-1)(k-1)$ and $k\ge 3$. We aim to determine the value of ar$(n,3,M_s)$ for $3s\le n<5s-2$ in this paper. Namely, we prove that if $3s<n<5s-2$ and $n$ is large enough, then ar$(n,3,M_s)=\text{ex}(n,3,M_{s-1})+2$. Here $\text{ex}(n,3,M_{s-1})$ is the Turán number of an $(s-1)$-matching. Thus this result confirms the conjecture of Özkahya and Young for $k=3$, $3s<n<5s-2$ and sufficiently large $n$. For $n=ks$ and $k\ge 3$, we present a new construction for the lower bound of $\text{ar}(n,k,M_s)$ which shows the conjecture by Özkahya and Young is not true. In particular, for $n=3s$, we prove that $\text{ar}(n,3,M_s)=\text{ex}(n,3,M_{s-1})+5$ for sufficiently large $n$.

21 pages