Non-Jumping Numbers for 4-Uniform Hypergraphs

Let r≥2 be an integer. A real number α ∈ 0,1) is a jump for r if for any Open image in new windowm, m≥r, any r-uniform graph with n>n₀( Open image in new windowm) vertices and at least Open image in new windowm vertices and at least Open image in new windowc=c(α) does not depend on Open image in new windowm. It follows from a theorem of Erd?s, Stone and Simonovits that every α ∈ 0,1) is a jump for r=2. Erd?s asked whether the same is true for r≥3. Frankl and Rödl gave a negative answer by showing that Open image in new windowr 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.