Definition 13 ($UV$-compression). Let $U,V\subset X$ with $|U|=|V|$, $U\cap V=\varnothing$ and $\max V>\max U$. We define the $UV$-compression as follows: for $A\subset X$,

$$C_{UV}(A)=\{\begin{array}{cc}A\cup U-V\hfill & \text{if }V\subset A\text{, }U\cap A=\varnothing \hfill \\ A\hfill & \text{otherwise}\hfill \end{array},$$

and for $A\subset X^{(r)}$, set

$$C_{UV}(A)=\{C_{UV}(A):A\in A\}\cup \{A\in A:C_{UV}\in A\}.$$