Line transversals in families of connected sets in the plane

Graph Theory Seminar
Tuesday, November 2, 2021 - 3:45pm for 1 hour (actually 50 minutes)
Shira Zerbib – Iowa State University – zerbib@iastate.edu
Anton Bernshteyn

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that under the same condition there are four lines intersecting all the sets. We also prove a colorful version of this result under weakened conditions on the sets, improving results of Holmsen from 2013. Our proofs use the topological KKM theorem. Joint with Daniel McGinnis.