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Imported statement

Let $k$ be a prime power, put $n=4k$, and let $\mathcal F$ be a family of $n/2$-element subsets of $[n]=\{1,\ldots,n\}$. If

$$|F\cap F'|\ne n/4 \qquad\text{for all distinct }F,F'\in\mathcal F,$$

then

$$|\mathcal F|\le 2\binom{n-1}{n/4-1}.$$

Kahn and Kalai state this as Theorem 2 on physical PDF p. 2 (journal p. 61) of arXiv v1. They attribute it to P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357--368, their reference [8].

This page records the exact external theorem interface used by the 1993 argument. The original Frankl--Wilson article and its proof were not independently checked in this source unit, so this is an explicit external premise rather than a reconstructed proof.

Application in the paper

In the equal-cut construction, $n=m=4k$. Choosing one side of each cut in a subfamily with diameter smaller than the full configuration produces a family of $m/2$-subsets with no intersection of size $m/4=k$. The displayed bound therefore limits every such subfamily to $2\binom{m-1}{m/4-1}$ cuts.

Bears on

• Problem 505: the bound is the combinatorial input to theorem_1 and remark_1; by itself it answers nothing about the problem.
• Problem 703: background only. The bound concerns families of $n/2$-element subsets of $[n]$ with intersection size $n/4$ forbidden, for $n=4k$ with $k$ a prime power. Such families are among those counted by the problem's $T(n,n/4)$, which ranges over families of arbitrary subsets, so the bound gives no upper bound on $T(n,r)$.