Using Lagrangians of hypergraphs to find non-jumping numbers(II)

Let r ≥ 2 be an integer. A real number α ∈ [0, 1) is a jump for r if for any ε > 0 and any integer m, m ≥ r, any r-uniform graph with n > n0(ε, m) vertices and density at least α + ε contains a subgraph with m vertices and density at least α + c, where c = c(α) does not depend on ε or m. It follows from a result of Erdős, Stone, and Simonovits 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-jumping numbers for r ≥ 3. However, there are a lot of unknowns on determining whether a number is a jump for r ≥ 3. In this paper, we first find two infinite sequences of non-jumping numbers for r = 4, then we extend one of the results to every r ≥ 4. Our approach is still based on the approach developed by Frankl and Rödl.