Cohomological Ramsey Theory

TL;DR

This paper links cohomology vanishing in Ramsey complexes to upper bounds on Ramsey numbers, extending Hom complex techniques from graph coloring to combinatorial Ramsey theory.

Contribution

It introduces Ramsey complexes as a generalization of Hom complexes and establishes a cohomological criterion for bounding Ramsey numbers.

Findings

Cohomology triviality implies coloring restrictions in complexes.

Explicit description of Ramsey complexes for upper bounds.

Extension of Hom complex theory to Ramsey problems.

Abstract

We show that the vanishing of certain cohomology groups of polyhedral complexes imply upper bounds on Ramsey numbers. Lovasz bounded the chromatic numbers of graphs using Hom complexes. Babson and Kozlov proved Lovasz conjecture and developed a Hom complex theory. We generalize the Hom complexes to Ramsey complexes. The main theorem states that if certain cohomology groups of the Ramsey complex Ram(dDelta_{p^k}, Sigma) are trivial, then the vertices of the simplicial complex Sigma cannot be n-colored such that every color correspond to a face of Sigma. In a corollary, we give an explicit description of the Ramsey complexes used for upper bounds on Ramsey numbers.