GENERATING NON-JUMPING NUMBERS OF HYPERGRAPHS

The concept of jump concerns the distribution of Turán densities. A number α ∈ [0, 1) is a jump for r if there exists a constant c > 0 such that if the Turán density of a family F of r-uniform graphs is greater than α, then the Turán density of F is at least α+c. To determine whether a number is a jump or non-jump has been a challenging problem in extremal hypergraph theory. In this paper, we give a way to generate non-jumps for hypergraphs. We show that if α, β are non-jumps for r1, r2 ≥ 2 respectively, then αβ(r1+r2)!r r1 1 r r2 2 r1!r2!(r1+r2) r1+r2 is a non-jump for r1 + r2. We also apply the Lagrangian method to determine the Turán density of the extension of the (r − 3)-fold enlargement of a 3-uniform matching.