given $s$ finite sets $a_1, \ldots, a_s$, determining the size of the union of the $s$ sets is an easy problem. determining the maximimum number of size $k$ subsets of an $n$ element set for which there does not exist $s$ sets which union has size $q$ is a very hard problem in general. many problems in extremal set theory can be restated in this language for particular choices of $s,k,q$. for instance, the case where $s=2$ is equivalent to the complete intersection theorem, and when $sk=q$, this is equivalent to the erd{\h o}s matching conjecture; one of the biggest open problems in the field. this talk is based off a recent paper of peter frankl and andrey kupavskii.