On non-strong jumping numbers and density structures of hypergraphs

Abstract: Estimating Turán 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 Turán densities. A number is a jump for -uniform graphs if there exists a constant such that for any family of -uniform graphs, if the Turán density of is greater than , then the Turán density of is at least . A fundamental result in extremal graph theory due to Erdős and Stone implies that every number in is a jump for graphs. Erdős also showed that every number in is a jump for -uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in for -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 Turán densities for hypergraphs better by finding more non-strong-jumps. [Copyright &y& Elsevier]