A note on the Structure of Turán Densities of Hypergraphs

摘要

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 Erd?s and Stone implies that every α ∈ [0, 1) is a jump for r = 2. Erd?s asked whether the same is true for r ≥ 3. Frankl and R?dl 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 R?dl.