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Source. Published p. 219, Theorem 2.2, with its set-up on pp. 218–219; an external input from Frankl–Rödl (1987), Theorem 1.16, printed p. 265. (canonical PDF).

Fix integers $r,k\ge1$ and a real $\lambda>0$. There is $\epsilon>0$, depending on these fixed parameters, with the following property; decreasing it keeps the property, so one may take $\epsilon<1$. Let $n\ge1$ and $l_0,\ldots,l_k\ge1$ be integers with $\sum_jl_j=n$. Write $\mathcal P(l_0,\ldots,l_k)$ for the ordered partitions $(A_0,\ldots,A_k)$ of $[n]$ with $|A_j|=l_j$.

Let $M=(m_{t_1\cdots t_r})$ be an array of nonnegative integers indexed by $(t_1,\ldots,t_r)\in\{0,\ldots,k\}^r$, satisfying

$$m_{t_1\cdots t_r}\ge\lambda n, \qquad \sum_{t_1,\ldots,t_r:\,t_j=i}m_{t_1\cdots t_r}=l_i \quad(1\le j\le r,\ 0\le i\le k).$$

Every $\mathcal K\subseteq\mathcal P(l_0,\ldots,l_k)$ with

$$|\mathcal K|\ge(1-\epsilon)^n\frac{n!}{l_0!\cdots l_k!}$$

contains partitions $A^{(1)},\ldots,A^{(r)}$ whose full joint intersections are exactly $M$:

$$|A^{(1)}_{t_1}\cap\cdots\cap A^{(r)}_{t_r}|=m_{t_1\cdots t_r}.$$

External proof scope. The full original proof and its same-paper prerequisites are now compiled at Frankl–Rödl (1987), Theorem 1.16. This remains an external input to the 2004 paper; the statement above records its exact imported form. The original statement was checked on printed p. 265 of the author-hosted PDF. The printed 1987 family threshold is the same weak inequality, at least $(1-\epsilon)^n$ times the multinomial count, as the one displayed here. For its strict cell lower bound, use $\lambda/2$; its positive relative pattern-count conclusion implies existence, since the full-family pattern count is positive by allocating disjoint blocks of the prescribed integer sizes. Only eventual dimensions are needed in the present application.

Pairwise intersections alone are not the input. The lower bound is required for every one of the $(k+1)^r$ joint cells. The constants are uniform as $n$ and the admissible integer arrays vary.

Source precision.

The published display suppresses the fixed $r,k$ dependence in $\epsilon(\lambda)$ and does not separately write that the target array has integer entries. Its condition (i) is printed for "any $0\le t_0,t_1,\ldots,t_r\le k$" [sic] (p. 219); the indices are $t_1,\ldots,t_r$. Integrality is necessary for any intersection-count conclusion. The quoted 2004 theorem uses the weak density endpoint.

Bears on. #174.