Minimal hypergraph non-jumps

TL;DR

This paper develops a general framework for the Frankl-R"odl method to identify non-jumps in hypergraph densities, providing new smaller non-jumps and showing the method's limitations.

Contribution

It introduces a general setting for the Frankl-R"odl method and applies it to find new, smaller non-jumps in hypergraph densities, highlighting the method's bounds.

Findings

Identified new non-jumps smaller than previously known.

Demonstrated the current method's limitations in proving smaller non-jumps.

Abstract

An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/|V(G)^{(r)}|$. We say $\alpha$ is a jump for $r$-graphs if there is some constant $\delta=\delta(\alpha)$ such that, for each $\varepsilon>0$ and $n\geq r$, any sufficiently large $r$-graph of density at least $\varepsilon$ has a subgraph of order $n$ and density at least $\alpha+\delta$. For $r=2$, all $\alpha$ are jumps. For $r\geq 3$, Erd\H{o}s showed all $[0,\frac{r!}{r^r})$ are jumps, and conjectured all $[0,1)$ are jumps. Since then, a variety of non-jumps have been proved, using a method introduced by Frankl and R\"odl. Our aim in this paper is to provide a general setting for this method. As an application, we give several new non-jumps, which are smaller than any previously known. We also demonstrate that these are the smallest the current method can prove.