Abstract
A hypergraph H is properly edge-colored if for every vertex v in H, all the edges incident to v have distinct colors. A rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. For a fixed r-uniform hypergraph F, the maximum number of edges in a properly edge-colored r-uniform hypergraph on n vertices without a rainbow copy of F is called the rainbow Turán number of F, denoted by ex⁎(n,r,F). In this paper, we determine ex⁎(n,3,F) when F is a matching of size s+1 for all n≥6.1s. Furthermore, we determine ex⁎(n,r,F) when F is a forest of hyperstars for sufficiently large n. Here, a hyperstar is an r-uniform hypergraph with all edges having at least one vertex in common.
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