UV-compression ($C_{U,V}$)

Let $U,V\subseteq [n]$ be disjoint and have the same cardinality. For a set $A$, its $UV$-compression isFor a uniform set family $\mathcal{A}$, a member is replaced only when its image is absent from $\mathcal{A}$; this convention preserves the family's cardinality.

Let $U,V$ be disjoint, equally sized, nonempty sets, with $\min U<\min V$. Suppose an intersecting family is unchanged by all such UV-compressions of smaller size. Then its $(U,V)$-UV-compression is still an intersecting family. Indeed, a newly moved member $A^{\prime }=(A\setminus V)\cup U$ can only be disjoint from a retained old member $B$, since two newly moved members share $U$. Here $B\cap U=\varnothing$ and $V^{\prime }=B\cap V\ne \varnothing$. If $V^{\prime }=V$, retaining $B$ forces $(B\setminus V)\cup U$ to have been present, contradicting the old intersection property with $A$. Otherwise choose $U^{\prime }\subset U$ with $|U^{\prime }|=|V^{\prime }|$ and $\min U\in U^{\prime }$. Stability under the smaller $(U^{\prime },V^{\prime })$-UV-compression forces $(A\setminus V^{\prime })\cup U^{\prime }$ to be present; this member is disjoint from $B$, again a contradiction. Repeatedly applying a changing UV-compression of smallest size terminates, since the sum of lexicographic order positions decreases. A terminal set family is a lexicographic initial segment: a missing earlier member $X$ and present later member $Y$ would be moved by $U=X\setminus Y$, $V=Y\setminus X$.

A set family is left-compressed when replacing a member $j$ by a missing smaller element $i<j$ always produces another member. Equivalently, it is fixed by every $\{i\},\{j\}$-compression with $i<j$.

Suppose that for every $x\in U$ there is a $y\in V$ such that $\mathcal{A}$ is fixed by the smaller compression $C_{U\setminus \{x\},V\setminus \{y\}}$. ThenIndeed, deleting an element outside $U$ from a newly compressed member gives the $UV$-compression of an old shadow member. If the deleted element is $x\in U$, stability under the chosen smaller compression shows that the resulting set already belongs to the old lower shadow. Thusand compression preserves cardinality.