Minimal hypergraph non-jumps

Abstract

An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/|V(G)^{(r)}|$. We say $α$ is a jump for $r$-graphs if there is some constant $δ=δ(α)$ 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 $α+δ$. For $r=2$, all $α$ are jumps. For $r\geq 3$, Erdős 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 Rödl. 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.