Using Lagrangians of Hypergraphs to Find Non-Jumping Numbers (I)

Abstract.Let r ≥ 2 be an integer. A real number $$\alpha \in [0, 1)$$ is a jump for r if for any $$\varepsilon > 0$$ and any integer m, m ≥ r, any r-uniform graph with $$n > n_0(\varepsilon ,m)$$ vertices and density at least $$\alpha + \varepsilon$$ contains a subgraph with m vertices and density at least α + c, where c = c(α) does not depend on $$\varepsilon$$ or m. It follows from a result of Erdős, Stone, and Simonovits that every $$\alpha \in [0, 1)$$ is a jump for r = 2. Erdőos 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.