Venn diagrams as forbidden hypergraph traces

We study the maximum size of a set system that contains no $k$-Venn diagram, denoted by $VD_k$, as a trace.

For every fixed $k\ge 3$, we prove $\text{ex}_{tr}(n,VD_k)=O_k(n^{2^k-2k+1})$, improving the direct Sauer-Shelah bound $O_k(n^{2^k-1})$.

In particular, for $k=4$ the exponent decreases from $15$ to $9$.

The proof starts from the theorem of Keevash, Leader, Long and Wagner for $VD_3$ and uses induction on $k$ in which two new Venn regions are forced for free at each added edge.

We also record lower-bound constructions for Venn diagrams in fixed uniformity, explicit bounds for the $4$-uniform $3$-Venn problem, and a fixed uniformity trace result for the loose triangle.