On non-strong jumping numbers and density structures of hypergraphs |
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Authors: | Yuejian Peng Cheng Zhao |
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Institution: | Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN, 47809, USA |
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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 α∈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 Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in 0,1) is a jump for graphs. Erd?s also showed that every number in 0,r!/rr) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in r!/rr,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 Turán densities for hypergraphs better by finding more non-strong-jumps. |
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Keywords: | Extremal problems in hypergraphs Turá n density Erd?s jumping constant conjecture |
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