A cornerstone result of extremal combinatorics due to Sauer and Shelah bounds the size of a family in terms of its VC-dimension. In this talk we will make progress towards establishing a dual version of the Sauer-Shelah lemma, where the roles of points and sets are reversed. This is equivalent to asking how many sets can we take without creating a $k$-Venn diagram, i.e., $k$ sets with all $2^k$ regions of the form $B_1 \cap B_2 \cap ... \cap B_k$, where $B_i \in \{A_i , [n] - A_i\}$, are non-empty. We will solve the first interesting case by showing that a family without a 3-Venn has size at most $O(n^3)$.
This is joint work with Peter Keevash, Imre Leader, and Jason Long.