11-doilies with vertex sets of sizes \(275, 286,\dots,462\) (Q1773843)

An \(n\)-Venn diagram consists of \(n\) simple closed curves in the plane with the property that all possible intersections of their interiors and exteriors are nonempty and connected. If the diagram is symmetric in the sense that it is not changed by successive rotations through an angle of \(2\pi/n\) about a fixed center, it is called an \(n\)-doily. It can be also viewed as a planar graph where vertices are the intersection points of the curves and edges correspond to the segments of curves with vertices as endpoints. The purpose of this paper is to show that 11-doilies exist for vertex sets of any size divisible by 11 between 275 and 462. This, together with previous results by the same authors, implies that 11-doilies exists for any possible size of the vertex set, thus answering a question of B.~Grünbaum.