Note on Upper Density of Quasi-Random Hypergraphs

In 1964, Erdős proved that for any α > 0, an l-uniform hypergraph G with n > n0(α, l) vertices and α ( n l ) edges contains a large complete l-equipartite subgraph. This implies that any sufficiently large G with density α > 0 contains a large subgraph with density at least l!/ll. In this note we study a similar problem for l-uniform hypergraphs Q with a weak quasi-random property (i.e. with edges uniformly distributed over the sufficiently large subsets of vertices). We prove that any sufficiently large quasi-random l-uniform hypergraph Q with density α > 0 contains a large subgraph with density at least (l−1)! ll−1−1 . In particular, for l = 3, any sufficiently large such Q contains a large subgraph with density at least 14 which is the best possible lower bound. We define jumps for quasi-random sequences of l-graphs and our result implies that every number between 0 and (l−1)! ll−1−1 is a jump for quasi-random l-graphs. For l = 3 this interval can be improved based on a recent result of Glebov, Král’ and Volec. We prove that every number between [0, 0.3192) is a jump for quasi-random 3-graphs.