Graphs and Combinatorics

, Volume 34, Issue 3, pp 489–499

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

Original Paper

## 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.

## Keywords

Extremal problems in hypergraphs Turán density Erdős jumping constant conjecture Lagrangians of uniform hypergraphs

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