  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.
