Let $r\ge 1$ and define $T(n,r)$ to be maximal such that there exists a family $\mathcal{F}$ of subsets of $\{1,\dots ,n\}$ of size $T(n,r)$ such that $|A\cap B|\ne r$ for all $A,B\in \mathcal{F}$.Estimate $T(n,r)$ for $r\ge 2$. In particular, is it true that for every $ϵ>0$ there exists $\delta >0$ such that for all $ϵn<r<(1/2-ϵ)n$ we have$$T(n,r)<(2-\delta {)}^n?$$