Non-jumping numbers for 5-uniform hypergraphs

摘要

Let ℓ and r be integers. 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 at least (α+ɛ)(nr)ɛ edges contains a subgraph with m vertices and at least (α+c)(mr) edges, where c=c(α) is positive and does not depend on ε and m. It follows from a theorem 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. However, Frankl and Rödl gave a negative answer by showing that 1−1ℓr−1 is not a jump for r if r ≥ 3 and ℓ > 2r. Peng gave more sequences of non-jumping numbers for r=4 and r ≥ 3. However, there are also a lot of unknowns on determining whether a number is a jump for r ≥ 3. Following a similar approach as that of Frankl and Rödl, we give several sequences of non-jumping numbers for r=5, and extend one of the results to every r ≥ 5, which generalize the above results.