AbstractWe formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle Pr1(μ+,μ+,μ+, cf (μ)) for singularμ.
An ergodic Markov chain is proved to be the realization of a random walk in a directed graph subject to a synchronizing road coloring. The result ensures the existence of appropriate random mappings in Propp-Wilson's coupling from the past. The proof is based on the road coloring theorem. A necessary and sufficient condition for approximate preservation of entropies is also given.
We find a non-invertible matrix representation for Van der Waerden's coloring theorem for two distinct colors in a one-dimensional periodic lattice. Using this, an infinite one-dimensional antiferromagnetic Ising system is mapped to a pseudo-ferromagnetic one, thereby relating the couplings. All this is reminiscent of renormalization group.
Hindmanʼs coloring theorem in arbitrary semigroups