A Note on Non-jumping Numbers for r-Uniform Hypergraphs

Abstract

A real number $\alpha \in [0,1)$ is a jump for an integer $r\ge 2$ if there exists a constant $c>0$ such that any number in $(\alpha , \alpha +c]$ cannot be the Turán density of a family of r-uniform graphs. Erdős and Stone showed that every number in [0,1) is a jump for $r=2$. Erdős asked whether the same is true for $r\ge 3$. Frankl and Rödl gave a negative answer by showing the existence of non-jumps for $r\ge 3$. Recently, Baber and Talbot showed that every number in $[0.2299,0.2316)\bigcup [0.2871, \frac{8}{27})$ is a jump for $r=3$ using Razborov’s flag algebra method. Pikhurko showed that the set of non-jumps for every $r\ge 3$ has cardinality of the continuum. But, there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we show that $1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}$ is a non-jump for $r\ge 4$ and $l\ge 3$ which generalizes some earlier results. We do not know whether the same result holds for $r=3$. In fact, when $r=3$ and $l=3$, $1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}={2 \over 9}$, and determining whether ${2 \over 9}$ is a jump or not for $r=3$ is perhaps the most important unknown question regarding this subject. Erdős offered $500 for answering this question.