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Symmetric 1-dependent colorings of the integers. (English) Zbl 1385.60014
Summary: In a recent paper, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary $$k$$-dependent $$q$$-coloring for any $$k$$ and $$q$$. When the analogous construction is carried out for $$q>4$$ colors, the resulting process is not $$k$$-dependent for any $$k$$. We construct here a process that is symmetric in the colors and $$1$$-dependent for every $$q\geq 4$$. The construction uses a recursion involving Chebyshev polynomials evaluated at $$\sqrt{q}/2$$.

##### MSC:
 60C05 Combinatorial probability 60G10 Stationary stochastic processes 05C15 Coloring of graphs and hypergraphs
##### Keywords:
random colorings; one-dependent processes
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