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

Abstract

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the anti-Ramsey number $ar(\mathcal{G},\mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ is the maximum number of colors needed to color the edges of $\mathcal{G}$ so that there does not exist a rainbow copy of $\mathcal{H}$. 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 $a{r}_r({\mathcal{K}}_{n_1,\dots ,n_r},M_k)$ of $k$-matchings in complete $r$-partite $r$-uniform hypergraphs and show the uniqueness of the extremal coloring. Also, we show that ${\mathcal{K}}_{k-1,n_2,\dots ,n_r}$ is the unique extremal hypergraph for Turán number $e{x}_r({\mathcal{K}}_{n_1,\dots ,n_r},M_k)$ and show that $a{r}_r({\mathcal{K}}_{n_1,\dots ,n_r},$ $M_k)=e{x}_r({\mathcal{K}}_{n_1,\dots ,n_r},M_{k-1})+1$, which gives a multi-partite version result of Özkahya and Young's conjecture.

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