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 hypergraph is rainbow if all its edges are 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 \(r\ge 3\) , which extends the results of Gu et al. (J Discret Math 34(1):271–307, 2020)
