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We find families of graphs $G$ and subgraphs $H$ of $G$ such that the numberof edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determinedby lattice point counts or a Hodge structure on the cohomology of a certaintoric variety. In general, this gives a class of ``anti-Ramsey theoryproblems'' with a geometric structure. For example, we find one for Ramseynumbers of classes of such graphs. The key observation is that our previousresult expressing simplicial chromatic polynomials in terms of $h$-vectors ofauxiliary simplicial complexes can be reinterpreted as one on edge colorings ofgraphs 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 forbiddensubgraphs where edge colorings avoiding monochromatic colorings of theforbidden subgraphs depend on lattice point counts or Hodge structures on thecohomology of toric varieties.

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