Anti-Ramsey number of matchings in hypergraphs

Abstract

A k-matching in a hypergraph is a set of k edges such that no two of these edges intersect. The anti-Ramsey number of a k-matching in a complete s-uniform hypergraph H on n vertices, denoted by ar(n,s,k), is the smallest integer c such that in any coloring of the edges of H with exactly c colors, there is a k-matching whose edges have distinct colors. The Turán number, denoted by ex(n,s,k), is the the maximum number of edges in an s-uniform hypergraph on n vertices with no k-matching. For k≥3, we conjecture that if n>sk, then ar(n,s,k)=ex(n,s,k−1)+2. Also, if n=sk, then ar(n,s,k)={ex(n,s,k−1)+2if k<csex(n,s,k−1)+s+1if k≥cs, where cs is a constant dependent on s. We prove this conjecture for k=2,k=3, and sufficiently large n, as well as provide upper and lower bounds.