On non-strong jumping numbers and density structures of hypergraphs

Estimating Turan densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of 'jump' concerns the distribution of Turan densities. A number @[email protected]?[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turan density of F is greater than @a, then the Turan density of F is at least @a+c. A fundamental result in extremal graph theory due to Erdos and Stone implies that every number in [0,1) is a jump for graphs. Erdos also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rodl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turan densities for hypergraphs better by finding more non-strong-jumps.