Anti-Ramsey theory problems, lattice point counts on polytopes, and Hodge structures on the cohomology of toric varieties

TL;DR

This paper explores the connection between anti-Ramsey problems, lattice point counts on polytopes, and Hodge structures on toric variety cohomology, revealing geometric structures underlying certain graph coloring problems.

Contribution

It introduces a novel geometric framework for anti-Ramsey problems using lattice point counts and Hodge structures, extending previous combinatorial results.

Findings

Edge colorings relate to lattice point counts in polytopes.

Hodge structures on toric varieties encode coloring avoidance properties.

New classes of graphs with coloring constraints are characterized geometrically.

Abstract

We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric variety. In general, this gives a class of ``anti-Ramsey theory problems'' with a geometric structure. For example, we find one for Ramsey numbers of classes of such graphs. The key observation is that our previous result expressing simplicial chromatic polynomials in terms of $h$-vectors of auxiliary simplicial complexes can be reinterpreted as one on edge colorings of graphs avoiding monochromatic colorings of specified forbidden subgraphs. Specializing to simplicial complexes arising from triangulations of polytopes (e.g. unimodular triangulations), we obtain families of graphs and forbidden subgraphs where edge colorings avoiding monochromatic colorings of the forbidden subgraphs depend on lattice point counts or Hodge structures on the cohomology of toric varieties.