families and the sizes of pairs of cross-intersecting families.+0.1cm

Both ${{\mathcal A}}$ and ${{\mathcal B}}$ are determined by their lexicographically last set. In this section, we use the lexicographical/containment order on $2^{[2,n]}$, which is defined as follows: $A\prec B$ iff $A\supset B$ or the minimal element of $A\setminus B$ is smaller than the minimal element of $B\setminus A$. Let us recall some notions and results from [@FK1] related to the Kruskal–Katona theorem and cross-intersecting families. For sets $S$ and $X$, $|S\cap X|\le a$, we define $$\mathcal L(X,S,a):=\Big\{A\in {X\choose a}\ :\ A\prec S\cap X\Big\}.$$ For example, the family $\{G\in{[2,n]\choose 10}\ :\  2\in G, G\cap\{3,4\}\ne \emptyset\}$ is the same as the family $\mathcal L([2,n],S,10)$ for $S=\{1,2,4\}$. If ${\mathcal G}=\mathcal L(X,S,a)$ for a certain set $S$, then we say that $S$ is the of ${\mathcal G}$.

Note that, in what follows, we shall work with $X=[2,n]$ and will omit $X$ from the notation for shorthand. For convenience, we shall assume that $1\in S$ (motivated by the fact that $S$ will stand for the characteristic set for the subfamily of all sets containing 1 in the original family), while $T\subset [2,n]$.

\[defrespa\] We say that two sets $S\subset [n]$ and $T\subset [2,n]$ , if 