Rado Numbers of Regular Nonhomogeneous Equations

TL;DR

This paper investigates Rado numbers for a class of regular nonhomogeneous equations, providing bounds and conditions for their values in terms of homogeneous cases, with specific examples illustrating the results.

Contribution

It introduces bounds and conditions for Rado numbers of nonhomogeneous equations based on homogeneous cases, advancing understanding of their combinatorial properties.

Findings

Upper bounds for Rado numbers are established.

Sufficient conditions for lower bounds are provided.

Exact Rado numbers are computed in specific cases.

Abstract

We consider Rado numbers of the regular equations $\mathcal{E}(b)$ of the form \[ c_1x_1+c_2x_2+\dots+ c_{k-1}x_{k-1} = x_k + b, \] where $b \in \mathbb{Z}$ and $c_i \in \mathbb{Z}^{+}$ for all $i$. We give the upper bounds and the sufficient condition for the lower bounds for $t$-color Rado numbers $r(\mathcal{E}(b);t)$ in terms of $r(\mathcal{E}(0);t)$ for both $b>0$ and $b<0$. We also give examples where the exact values of Rado numbers are obtained from these results.