Symmetric 1-dependent colorings of the integers

Abstract

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 $\sqrt{q}/2$.