Subgraph densities in hypergraphs

Let r ≥ 2 be an integer. A real number α ∈ [0, 1) is a jump for r if for any 2 > 0 and any integer m ≥ r, any r-uniform graph with n > n0(2, m) vertices and density at least α + 2 contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on 2 and m. A result of Erdős, Stone and Simonovits 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.