Introducing shift-constrained Rado numbers
Brian Hopkins · 2022
The 2-color Rado number for ax + by = z with positive integers 1 ≤ a ≤ b is known; this is the least integer such that any 2-coloring of the integers from 1 to the Rado number must include a solution to the equation consisting of numbers that have been assigned the same color. We modify the requirements by introducing constraints on the colorings. These constraints are motivated by symbolic dynamics, specifically the golden mean shift and a version of the even shift, both over a 2-letter alphabet. We establish several initial results, offer some conjectures, and outline possible directions for further research in this new study of shift-constrained Rado numbers.