On Jumping Densities of Hypergraphs

Abstract

A number ${\alpha\in [0, 1)}$ is a jump for an integer r ≥ 2 if there exists a constant c > 0 such that for any family ${{\mathcal F}}$ of r-uniform graphs, if the Turán density of ${{\mathcal F}}$ is greater than α, then the Turán density of ${{\mathcal F}}$ is at least α + c. A fundamental result in extremal graph theory due to Erdős and Stone implies that every number in [0, 1) is a jump for r = 2. Erdős also showed that every number in [0, r!/r ^r) is a jump for r ≥ 3. However, not every number in [0, 1) is a jump for r ≥ 3. In fact, Frankl and Rödl showed the existence of non-jumps for r ≥ 3. By a similar approach, more non-jumps were found for some r ≥ 3 recently. But there are still a lot of unknowns regarding jumps for hypergraphs. In this note, we show that if ${c\cdot{\frac{r!}{r^r}}}$ is a non-jump for r ≥ 3, then for every p ≥ r, ${c\cdot{\frac{p!}{p^p}}}$ is a non-jump for p.