Anti-Ramsey number of disjoint union of star-like hypergraphs

Given an r-graph (or r-uniform hypergraph) F, the anti-Ramsey number ar(n, r, F) is the minimum number c of colors such that for any edge-coloring of the complete r-graph Knr on n vertices with at least c colors, there is a subgraph F of Knr whose edges have distinct colors. Let S(r) 3 be the linear star of size three in r-graphs. In this paper, we obtain the exact anti-Ramsey number ar(n, r, S(r) is star-like if all of its edges share at least one common vertex. Moreover, let F be an r-graph which is a vertex-disjoint union of k +1 star-like r-graphs F0, F1, ... , Fk, where F0 = S(r) 3 and each Fi (i =1, ... , k) contains a subgraph isomorphic F0. We prove that for all r >= 3, k >= 1 and sufficiently large n, ar(n, r, F) = (n ) - (n-k) + ar(n - k, r, S(r)(c) 2023 Published by Elsevier B.V.