Minimal hypergraph non-jumps

An $r$-uniform hypergraph, or $r$-graph, has density $\mathrm{\mid }E(G)\mathrm{\mid }\mathrm{/}\mathrm{\mid }V(G{)}^{(r)}\mathrm{\mid }$. We say $\alpha$ is a jump for $r$-graphs if there is some constant $\delta =\delta (\alpha )$ such that, for each $\epsilon >0$ and $n\ge r$, any sufficiently large $r$-graph of density at least $\epsilon$ has a subgraph of order $n$ and density at least $\alpha +\delta$. For $r=2$, all $\alpha$ are jumps. For $r\ge 3$, 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.