Helly type theorems for open connected sets in the plane

Helly's theorem is a classical result in combinatorial geometry about the intersection patterns of convex sets in Euclidean spaces. For the 2-dimensional case, it asserts that for a family of convex sets in the plane, if every 3 members have a point in common, then all members in the family have a point in common. The topological Helly theorem, also proved by Helly, shows that the same assertion holds for a family of open and contractible sets where each intersection is again open and contractible.

In this talk, I will present Helly-type theorems for a more general set-system: a family of open connected sets in the plane where any intersection is again open and connected. In this setting, if every 4 members have a point in common, then all members in the family have a point in common. I will also discuss a fractional generalization of this.