2021/07/04 by Zhou, Hongyi
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2107.01550
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to have a monochromatic solution whenever the positive integers are finitely colored. In this paper, we provide a variant of this theorem. For k ≥ 2, we present conditions such that, when the set of variables is partitioned into k subsets, there is a solution such that the variables of each subset are monochromatic, which we call a semi-monochromatic solution. We adopt the smod p coloring by Graham, Rothschild, and Spencer but turn the existence of semi-monochromatic solution into the existence of common roots of linear polynomials. With this idea, one can further generalize the theorem to systems of linear equations over general algebraic number fields.