2026/07/19 by Adam Džavoronok, Tymofii Reizin, Jakub Šošovička
#math.CO
We study the maximum size of a set system that contains no k-Venn diagram, denoted by VDk, as a trace. For every fixed k≥ 3, we prove extr(n,VDk)=Ok(n2k-2k+1), improving the direct Sauer-Shelah bound Ok(n2k-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 VD3 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.