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

For a fixed family of r-uniform hypergraphs F, the anti-Ramsey number of F, denoted by ar (n, r, 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 F. Here, a rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. Let Pₖ and Cₖ 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, Pₖ) and ar (n, r, Cₖ) for all k 4 and r 3.