Non-Jumping Numbers for 4-Uniform Hypergraphs

Abstract

Let r≥2 be an integer. A real number α ∈ [0,1) is a jump for r if for any >0 and any integer m, m≥r, any r-uniform graph with n>n₀(,m) vertices and at least edges contains a subgraph with m vertices and at least edges, where c=c(α) does not depend on 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. Frankl and Rödl gave a negative answer by showing that is not a jump for r if r≥3 and l>2r. Following a similar approach, we give several sequences of non-jumping numbers generalizing the above result for r=4.