Symmetric 1-Dependent Colorings of the Integers

TL;DR

The paper introduces a new symmetric 1-dependent q-coloring of the integers for all q≥4, extending previous work and utilizing Chebyshev polynomials in the construction.

Contribution

It presents a novel recursive construction for symmetric 1-dependent colorings applicable to all q≥4, expanding the class of such processes.

Findings

Constructed symmetric 1-dependent q-colorings for all q≥4

Utilized Chebyshev polynomials in the recursive process

Extended previous results on stationary colorings of integers

Abstract

In a recent paper by the same authors, 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>=4. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.