Co-degree density of hypergraphs

For an r-graph H, let C(H)=min"Sd(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 [email protected]?{v} is an edge of H. Given a family F of r-graphs, the co-degree Turan 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 [email protected](F)[email protected]?supn->~co-ex(n,F)n. When r>=3, non-zero values of @c(F) are known for very few finite r-graphs families F. Nevertheless, our main result implies that the possible values of @c(F) form a dense set in [0,1). The corresponding problem in terms of the classical Turan density is an old question of Erdos (the jump constant conjecture), which was partially answered by Frankl and Rodl [P. Frankl, V. Rodl, Hypergraphs do not jump, Combinatorica 4 (2-3) (1984) 149-159]. We also prove the existence, by explicit construction, of finite F satisfying 0<@c(F)