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Source. Published p. 263, Theorem 1.9, and p. 274 (PDF).

Statement. For each fixed integer $r\ge2$, there are $\epsilon_r,\sigma_r>0$ and $n_r$ such that, if $n\ge n_r$, $|\mathcal F|\ge2^ne^{-\epsilon_rn}$ and $|l-n/4|\le\sigma_rn$, then $\mathcal F$ contains distinct members $F_1,\ldots,F_r$ with $|F_i\cap F_j|=l$ for every $i\ne j$. Only the intersection cardinalities must agree; their actual sets need not coincide.

Proof. For $r=2$, Theorem 1.1 supplies the assertion for a fixed small window around $n/4$. Suppose it holds for $r$. Apply Theorem 1.7 with output loss $\gamma<\epsilon_r/4$, and choose the new input loss $\epsilon_{r+1}<\epsilon_r/4$ small enough for that theorem. Choose the new window inside both its window and the inductive window. It gives at least $|\mathcal F|^2e^{-\gamma n}$ ordered target pairs. Thus some $F_0\in\mathcal F$ has a neighbor family

$$\mathcal N=\{F\in\mathcal F:|F\cap F_0|=l\}$$

of size at least $|\mathcal F|e^{-\gamma n}$. Remove $F_0$ itself if present. For sufficiently large $n$ the remaining size is still at least $2^ne^{-\epsilon_rn}$, since $\epsilon_{r+1}+\gamma<\epsilon_r$. The induction gives $r$ distinct members of this family with all mutual intersection sizes $l$. Together with $F_0$ they give the required $r+1$ members. Increase $n_{r+1}$ to cover all thresholds used. $\square$

Scope. The eventual threshold is explicit. The source introduction omits it, but its proof uses Theorem 1.7 in its large-$n$ range. For arbitrary fixed $r$, an assertion for every $n$ would even allow a full cube with fewer than $r$ members. The removal of $F_0$ handles the possible diagonal pair without assuming its size differs from $l$.

Dependencies. theorem_1_1, theorem_1_7.