A note on the uniformity threshold for Berge hypergraphs

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.

The Turán number of Berge book hypergraphs

“…The main reason to distinguish heavy edges is that whenever we have a subgraph of the shadow graph of H, and we want to show that it is the core of a Berge copy, we can greedily pick for its heavy edges new hyperedges containing a them, even if we have already picked some hyperedges to represent other edges. We will use a form of this observation from [9].…”

“…Let us consider now uniformity r. Gerbner [9] showed that if a graph F is a subgraph of a blow-up of another graph H and r ≥ |V (F )|, then for any Berge-F -free r-uniform n-vertex hypergraph H there is a set S of o(n 2 ) edges in the shadow graph of H such that every copy of H in the shadow graph contains an edge from S. Here we need to extend this to smaller values of r. Fortunately, the assumption r ≥ |V (F )| was used in [9] only to obtain that H has O(n 2 ) hyperedges (due to a result from [12]). This conclusion holds when F = F t in every uniformity by Proposition 2.6.…”

“…This conclusion holds when F = F t in every uniformity by Proposition 2.6. Thus most of the proof of the following statement can be found in [9]. We provide a proof for sake of completeness.…”

“…We also consider larger uniformity. Gerbner [9] proved that if r ≥ 2t+3, then ex r (n, Berge-B t ) = o(n 2 ). For other values of r, we determine ex r (n, Berge-B t ) asymptotically.…”

“…Our methods also work for the t-fan F t , which consists of t triangles sharing a vertex. Gerbner [9] proved that if r ≥ 4t + 2, then ex r (n, Berge-F t ) = o(n 2 ).…”