New bounds on the size-Ramsey numbers of hypergraphs

Abstract

We obtain two new bounds for the size-Ramsey numbers of k -uniform hypergraphs. First, we prove that R ˆ ( K n ( k ) ) ≥ R ˆ ( K n − 2 ( k − 2 ) ) ( k 2 ) ( R ( K n ( k ) ) 2 ) for k ≥ 4 and n ≥ k , which extends a result of Dudek, La Fleur, Mubayi and Rödl (2017). Second, extending the result of Tikhomirov (2024) to hypergraphs, we obtain that there is a positive constant c = c ( k ) such that for every n there is a k -uniform hypergraph G of order at most n with maximum degree k + 1 such that R ˆ ( G ) = Ω ( n exp ⁡ ( c log ⁡ n ) ) . This improves the previous lower bound R ˆ ( G ) = Ω ( n ( log ⁡ n ) c ) due to Dudek, La Fleur, Mubayi and Rödl in the same paper.