SOLVED - $250 Let$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 $T(n,r)<(2-\delta)^n?$ It is trivial that$T(n,0)=2^{n-1}$. Frankl and Füredi [FrFu84] proved that, for fixed$r$and$n$sufficiently large in terms of$r$, the maximal$T(n,r)$is achieved by taking $\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\rvert> \frac{n+r}{2}\textrm{ or }\lvert A\rvert < r\right\}$ when$n+r$is odd, and $\mathcal{F} = \left\{ A\subseteq \{1,\ldots,n\} : \lvert A\backslash \{1\}\rvert\geq \frac{n+r}{2}\textrm{ or }\lvert A\rvert < r\right\}$ when$n+r$is even. (Frankl [Fr77b] had earlier proved this for$r=1$and all$n$.) An affirmative answer to the second question implies that the chromatic number of the unit distance graph in$\mathbb{R}^n$(with two points joined by an edge if the distance between them is$1$) grows exponentially in$n\$, which was proved by alternative methods by Frankl and Wilson [FrWi81] - see [704].

The answer to the second question is yes, proved by Frankl and Rödl [FrRo87].