It is trivial that $T(n,0)=2^{n-1}$. Frankl and Füredi [FrFu84b] 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].

See also [702].

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This page was last edited 16 October 2025.