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 Turan 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 Rodl 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.