On the k-uniform set-coloring Ramsey numbers of hypergraphs

For k -uniform hypergraphs G 1 , … , G r and integers r > s ≥ 1 , the k -uniform set-coloring Ramsey number R k ( G 1 , … , G r ; r , s ) is the smallest integer N such that every coloring of the edges of complete k -uniform hypergraphs K N ( k ) with s -element subsets of [ r ] contains a copy of G i for some i ∈ [ r ] such that every edge contains the color element i . In the diagonal case G 1 = ⋯ = G r = K n ( k ) , we write this number as R k ( n ; r , s ) . Conlon, Fox, He, Mubayi, Suk and Verstraëte asked whether R 3 ( n ; 5,2 ) is double exponential in n . In this paper, we prove that 2 2 ( 1 / 2 − o ( 1 ) ) ( log 2 n ) 2 ≤ R 3 ( n ; 5,2 ) ≤ 2 2 O ( n ) . In addition, using the probabilistic method and the Lovász local lemma, we establish lower bounds for set-coloring Ramsey numbers of general k -uniform hypergraphs.