Anti-Ramsey numbers of loose paths and cycles in uniform hypergraphs

Abstract: For a fixed family of $r$-uniform hypergraphs $\mathcal{F}$, the anti-Ramsey number of $\mathcal{F}$, denoted by $ ar(n,r,\mathcal{F})$, is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-uniform hypergraph on $n$ vertices with at least $c$ colors, there is a rainbow copy of some hypergraph in $\mathcal{F}$. Here, a rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. Let $\mathcal{P}_k$ and $\mathcal{C}_k$ be the families of loose paths and loose cycles with $k$ edges in an $r$-uniform hypergraph, respectively. In this paper, we determine the exact values of $ ar(n,r,\mathcal{P}_k)$ and $ ar(n,r,\mathcal{C}_k)$ for all $k\geq 4$ and $r\geq 3$.