Hypergraphs do not jump

AbstractThe number α, 0≦α≦1, is a jump forr if for any positive ε and any integerm,m≧r, anyr-uniform hypergraph withn>no (ε,m) vertices and at least (α+ε) $$\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)$$ edges contains a subgraph withm vertices and at least (α+c) $$\left( {\begin{array}{*{20}c} m \\ r \\ \end{array} } \right)$$ edges, wherec=c(α) does not depend on ε andm. It follows from a theorem of Erdös, Stone and Simonovits that forr=2 every α is a jump. Erdös asked whether the same is true forr≧3. He offered $ 1000 for answering this question. In this paper we give a negative answer by showing that $$1 - \frac{1}{{l^{r - 1} }}$$ is not a jump ifr≧3,l>2r.