Non-jumps of hypergraphs

TL;DR

This paper introduces a method using patterns to identify non-jumps in 3-uniform hypergraphs, providing new examples that challenge the conjecture that all densities are jumps.

Contribution

It develops a novel pattern-based approach to find non-jumps in 3-uniform hypergraphs, expanding the known set of non-jump densities.

Findings

Identified new non-jumps for r=3 using the pattern method.

Provided a systematic approach to find non-jumps.

Challenged the conjecture that all densities are jumps.

Abstract

A density $\alpha\in [0, 1)$ is a jump for $r$ if there is some $c >0$ such that there does not exist a family of $r$-uniform hypergraphs $\mathcal{F}$ with Tur\'an density $\pi(\mathcal{F})$ in $(\alpha, \alpha + c)$. Erd\"os conjectured that all $\alpha\in [0, 1)$ are jumps for any $r$. This was disproven by Frankl and R\"odl when they provided examples of non-jumps. In this paper, we provide a method for finding non-jumps for $r = 3$ using patterns. As a direct consequence, we find a few more examples of non-jumps for $r = 3$.