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 $k\ge 4$ and $r\ge 3$.