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.

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.