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Source. Published pp. 259–267 (PDF). These definitions also fix conventions in the expanded proofs.

Ambient sizes in theorem statements are positive integers. A terminal empty ground set in a deletion proof is handled separately.

For an explicitly specified finite set $X$, write $\Omega(X;k)=\{A\subseteq X:|A|=k\}$ and $\Omega(X;\mathbf l)$ for ordered partitions of $X$ with cell sizes $\mathbf l=(l_1,\ldots,l_s)$. Its cardinality is $|X|!/\prod l_i!$. A family has no repeated members. Pairs and tuples in counting statements are ordered; no distinctness is imposed unless stated or forced by the pattern.

For $0<p<1$, define

$$\mu_{p,X}(\mathcal F)=\sum_{F\in\mathcal F} p^{|F|}(1-p)^{|X|-|F|},\qquad d_X(\mathcal F)=\mu_{1/2,X}(\mathcal F)=|\mathcal F|/2^{|X|}.$$

The ground set is retained even if its points occur in no member. This is necessary for the slice identities used in the proofs. The source's notation $p(\mathcal F)$ is written using $|\bigcup\mathcal F|$ on p. 267; we use the explicit current ambient ground set throughout the deletion argument, rather than silently removing unused coordinates.

For $x\in X$, put $\mathcal F_0=\{F\in\mathcal F:x\notin F\}$ and $\mathcal F_1=\{F\setminus\{x\}:x\in F\in\mathcal F\}$, both on $X\setminus\{x\}$. Thus $\mu_p(\mathcal F)=(1-p)\mu_p(\mathcal F_0)+p\mu_p(\mathcal F_1)$. Write $(\mathcal F,\mathcal G)\in\mathcal P(X;[a,b])$ if every cross intersection avoids every integer in $[a,b]$. Directly from whether $x$ is present,

$$\begin{aligned} (\mathcal F_1,\mathcal G_1)&\in\mathcal P(X-x;[a-1,b-1]),\\ (\mathcal F_0,\mathcal G_0\cup\mathcal G_1)&\in\mathcal P(X-x;[a,b]),\\ (\mathcal F_1,\mathcal G_0\cap\mathcal G_1)&\in\mathcal P(X-x;[a-1,b]). \end{aligned}$$

For partition families $\mathcal A^{(i)}$, the joint pattern is the array

$$m_{j_1\ldots j_r} =\left|\bigcap_{i=1}^r A^{(i)}_{j_i}\right|.$$

It is compatible with the specified cell sizes if its entries are nonnegative integers and its one-coordinate marginals are those sizes. Compatibility is equivalent to realizability: partition $X$ into labeled atoms of the given sizes, then take the indicated unions. Consequently, the number of full-family tuples realizing a compatible pattern is exactly

$$N(M)=\frac{n!}{\prod_{j_1,\ldots,j_r}m_{j_1\ldots j_r}!}.$$

The notation $i_M$ counts such tuples in the chosen families. For two uniform set families, $i_l$ counts pairs with intersection size $l$; the full-family count is

$$N(n;k,h,l)=\binom nk\binom kl\binom{n-k}{h-l}.$$

The full proofs often use a density $e^{-\epsilon n}$ in place of $(1-\epsilon')^n$, where $\epsilon'=1-e^{-\epsilon}$. Statements using these two conventions are equivalent after renaming the positive constant. All auxiliary proportions rounded to integers are assigned explicitly; parameters written as cell sizes are always integers.