Anti-Ramsey number of matchings in r-partite r-uniform hypergraphs

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G, the anti-Ramsey number ar(G, H) of H in G is the maximum number of colors in a coloring of the edges of G so that there does not exist a rainbow copy of H. Li et al. determined the anti-Ramsey number of k-matchings in complete bipartite graphs. Jin and Zang showed the uniqueness of the extremal coloring. In this paper, as a generalization of these results, we determine the anti-Ramsey number ar(r)(K-n1,K-...,K-nr, M-k) of k-matchings in complete r-partite r-uniform hypergraphs and show the uniqueness of the extremal coloring. Also, we show that Kk-1,n2, ,nr is the unique extremal hypergraph for Turan number ar(r)(K-n1,K-...,K-nr, M-k) and show that ar(r)(K-n1,K-...,K-nr, M-k) = ex(r)(K-n1,K-..,K-nr, Mk-1) + 1, which gives a multi-partite version result of Ozkahya and Young's conjecture. (C)& nbsp;2021 Published by Elsevier B.V.