Abstract
Let r ≥ 2 be an integer. A real number α ∈ [0, 1) is a jump for r if there exists c > 0 such that no number in (α, α + c) can be the Turán density of a family of r-uniform graphs. A result of Erdős and Stone implies that every α ∈ [0, 1) is a jump for r = 2. Erdős asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.
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Received: October 18, 2006. Final Version received: November 9, 2007.
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Peng, Y. A note on the Structure of Turán Densities of Hypergraphs. Graphs and Combinatorics 24, 113–125 (2008). https://doi.org/10.1007/s00373-008-0773-0
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DOI: https://doi.org/10.1007/s00373-008-0773-0