Anti-Ramsey number of matchings in hypergraphs

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 > s k , then ar ( n , s , k ) = ex ( n , s , k − 1 ) + 2 . Also, if n = s k , then ar ( n , s , k ) = { ex ( n , s , k − 1 ) + 2 if  k < c s ex ( n , s , k − 1 ) + s + 1 if  k ≥ c s , where c s 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.