A note on the uniformity threshold for Berge hypergraphs. (October 2022) - Gerbner, Dániel
Abstract: A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Grósz, Methuku and Tompkins in 2020 showed that for any graph F, there is an integer r 0 = r 0 ( F ), such that for any r ≥ r 0, any r -uniform hypergraph without a Berge copy of F has o ( n 2 ) hyperedges. The smallest such r 0 is called the uniformity threshold of F and is denoted by th ( F ) . They showed that th ( F ) ≤ R ( F, F ′ ), where R denotes the off-diagonal Ramsey number and F ′ is any graph obtained form F by deleting an edge. We improve this bound to th ( F ) ≤ R ( K χ ( F ), F ′ ), and use the new bound to determine th ( F ) exactly for several classes of graphs.