Co-degree density of hypergraphs

For an r-graph H, let C(H) = minS 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 supn→∞ 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 [14]. We also prove the existence, by explicit construction, of finite F satisfying 0 < γ(F) < minF∈F γ(F ). This is parallel to recent results on the Turán density by Balogh [1], and by the first author and Pikhurko [23].