Uniformity thresholds for the asymptotic size of extremal Berge-F-free hypergraphs

Abstract Let F = ( U , E ) be a graph and H = ( V , E ) be a hypergraph. We say that H contains a Berge- F if there exist injections ψ : U → V and φ : E → E such that for every e = { u , v } ∈ E , { ψ ( u ) , ψ ( v ) } ⊂ φ ( e ) . Let ex r ( n , F ) denote the maximum number of hyperedges in an r -uniform hypergraph on n vertices which does not contain a Berge- F . For small enough r and non-bipartite F , ex r ( n , F ) = Ω ( n 2 ) ; we show that for sufficiently large r , ex r ( n , F ) = o ( n 2 ) . Let th ( F ) = min { r 0 : ex r ( n , F ) = o ( n 2 ) for all r ≥ r 0 } . We show lower and upper bounds for th ( F ) , the uniformity threshold of F . In particular, we obtain that th ( △ ) = 5 , improving a result of Győri (2006). We also study the analogous problem for linear hypergraphs. Let ex r L ( n , F ) denote the maximum number of hyperedges in an r -uniform linear hypergraph on n vertices which does not contain a Berge- F , and let the linear uniformity threshold th L ( F ) = min { r 0 : ex r L ( n , F ) = o ( n 2 ) for all r ≥ r 0 } . We show that th L ( F ) is equal to the chromatic number of F .