Generating non-jumps from a known one

Abstract

Let $r\ge 2$ be an integer. The real number $\alpha \in [0,1]$ is a jump for $r$ if there exists a constant $c>0$ such that for any $\epsilon >0$ and any integer $m\ge r$, there exists an integer $n_0(\epsilon ,m)$ satisfying any $r$-uniform graph with $n\ge n_0(\epsilon ,m)$ vertices and density at least $\alpha +\epsilon$ contains a subgraph with $m$ vertices and density at least $\alpha +c$. A result of Erdős, Stone and Simonovits implies that every $\alpha \in [0,1)$ 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 that $1-\frac{1}{l^{r-1}}$ is not a jump for $r$ if $r\ge 3$ and $l>2r$. After that, more non-jumps are found using a method of Frankl and Rödl. In this note, we show a method to construct maps $f:[0,1]\to [0,1]$ that preserve non-jumps, if $\alpha$ is a non-jump for $r$ given by the method of Frankl and Rödl, then $f(\alpha )$ is also a non-jump for $r$. We use these maps to study hypergraph Turán densities and answer a question posed by Grosu.