Georgia Institute of Technology College of Sciences

Researchers here at Georgia Tech initiated a "Ramsey Theory" on binary trees and used the resulting tools to show that the local dimension of a poset is not bounded in terms of the tree-width of its cover graph. Subsequently, in collaboration with colleagues in Germany and Poland, we extended these Ramsey theoretic tools to solve a problem posed by Seymour. In particular, we showed that there is an infinite sequence of graphs with bounded tree-chromatic number and unbounded path-chromatic number. An interesting detail is that our research showed that a family conjectured by Seymour to have this property did not. However, the insights gained in this work pointed out how an appropriate modification worked as intended. The Atlanta team consists of Fidel Barrera-Cruz, Heather Smith, Libby Taylor and Tom Trotter The European colleagues are Stefan Felsner, Tamas Meszaros, and Piotr Micek.