Abstract
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 arr(Kn1,…,nr,Mk) 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 Turán number exr(Kn1,…,nr,Mk) and show that arr(Kn1,…,nr, Mk)=exr(Kn1,…,nr,Mk−1)+1, which gives a multi-partite version result of Özkahya and Young's conjecture.
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