Graph with any rational density and no rich subsets of linear size

A well-known application of the dependent random choice asserts that any n-vertex graph G with positive edge density contains a `rich' vertex subset U of size n¹⁻ᵒ⁽¹⁾ such that every pair of vertices in U has at least n¹⁻ᵒ⁽¹⁾ common neighbors. In 2003, using a beautiful construction on hypercube, Kostochka and Sudakov showed that this is tight: one cannot remove the $o(1)$ terms even if the edge density of G is $1/2$. In this paper, we generalize their result from pairs to tuples. To be precise, we show that given every pair of positive integers $p<q$, there is an n-vertex graph G for all sufficiently large n with edge density $p/q$ such that any vertex subset U of size Ω(n) contains q vertices, any $p+1$ of which have $o(n)$ common neighbors. The edge density $p/q$ is best possible. Our construction uses isoperimetry and concentration of measure on high dimensional complex spheres.