Co-degree density of hypergraphs

Abstract

For an r-graph H, let C ( H ) = min S d ( S ) , where the minimum is taken over all ( r − 1 ) -sets of vertices of H, and d ( S ) is the number of vertices v such that S ∪ { v } is an edge of H. Given a family F of r-graphs, the co-degree Turán number co - ex ( n , F ) is the maximum of C ( H ) among all r-graphs H which contain no member of F as a subhypergraph. Define the co-degree density of a family F to be γ ( F ) = lim sup n → ∞ co - ex ( n , F ) n . When r ⩾ 3 , non-zero values of γ ( F ) are known for very few finite r-graphs families F . Nevertheless, our main result implies that the possible values of γ ( F ) form a dense set in [ 0 , 1 ) . The corresponding problem in terms of the classical Turán density is an old question of Erdős (the jump constant conjecture), which was partially answered by Frankl and Rödl [P. Frankl, V. Rödl, Hypergraphs do not jump, Combinatorica 4 (2–3) (1984) 149–159]. We also prove the existence, by explicit construction, of finite F satisfying 0 < γ ( F ) < min F ∈ F γ ( F ) . This is parallel to recent results on the Turán density by Balogh [J. Balogh, The Turán density of triple systems is not principal, J. Combin. Theory Ser. A 100 (1) (2002) 176–180], and by the first author and Pikhurko [D. Mubayi, O. Pikhurko, Constructions of non-principal families in extremal hypergraph theory, Discrete Math. (Special issue in honor of Miklos Simonovits' 60th birthday), in press].