A note on the uniformity threshold for Berge hypergraphs. (October 2022)

• Record Type: Journal Article
• Authors: Gerbner, Dániel
• Is Part Of: European journal of combinatorics. Volume 105(2022)
• Publication Date: 2022-10
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2022.103561
• Languages: English
• ISSNs: 0195-6698

Abstract

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.