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

Abstract: Let $n,s,$ and $k$ be positive integers such that $k\geq 3$, $s\geq 3$ and $n\geq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\textrm{ar}(n,k,M_s)$ 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, \"Ozkahya and Young proposed a conjecture on the exact value of ar$(n,k,M_s)$ for all $n \geq sk$ and $k \geq 3$. A 2019 result by Frankl and Kupavskii verified this conjecture for all $n \geq sk+(s-1)(k-1)$ and $k \geq 3$. We aim to determine the value of ar$(n,3,M_s)$ for $3s \leq n < 5s-2$ in this paper. Namely, we prove that if $3s<n<5s-2$ and $n$ is large enough, then ar$(n,3,M_s)=\textrm{ex}(n,3,M_{s-1})+2$. Here $\textrm{ex}(n,3,M_{s-1})$ is the Tur\'an number of an $(s-1)$-matching. Thus this result confirms the conjecture of \"Ozkahya and Young for $k=3$, $3s<n<5s-2$ and sufficiently large $n$. For $n=ks$ and $k\geq 3$, we present a new construction for the lower bound of $\textrm{ar}(n,k,M_{s})$ which shows the conjecture by \"Ozkahya and Young is not true. In particular, for $n=3s$, we prove that $\textrm{ar}(n,3,M_s)=\textrm{ex}(n,3,M_{s-1})+5$ for sufficiently large $n$.