Hypergraphs do not jump

Abstract

The number α, 0≦α≦1, is a jump forr if for any positive ε and any integerm,m≧r, anyr-uniform hypergraph withn>n _o (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.