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

Abstract

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 S3(r) be the linear star of size three in r-graphs. In this paper, we obtain the exact anti-Ramsey number ar(n,r,S3(r)) for all r≥3 and sufficiently large n. An r-graph 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=S3(r) 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)=(nr)−(n−kr)+ar(n−k,r,S3(r)).