Positive co-degree densities and jumps

The minimum positive co-degree of a nonempty r -graph H , denoted by δ r - 1 + ( H ) , is the largest integer k such that for every ( r - 1 ) -set S ⊂ V ( H ) , if S is contained in a hyperedge of H , then S is contained in at least k hyperedges of H . Given a family ℱ of r -graphs, the positive co-degree Turán function co + ex ( n , ℱ ) is the maximum of δ r - 1 + ( H ) over all n -vertex r -graphs H containing no member of ℱ . The positive co-degree density of ℱ is γ + ( ℱ ) = lim n → ∞ co + ex ( n , ℱ ) n . While the existence of γ + ( ℱ ) is proved for all families ℱ , only few positive co-degree densities are known exactly. For a fixed r ≥ 2 , we call α ∈ 0 , 1 an achievable value if there exists