The Discrete Rado Number for $x_1 + x_2 + \dots + x_m + c = 2x_0$

TL;DR

This paper determines explicit formulas for the smallest integers ensuring monochromatic solutions to specific linear equations under 2-colorings, revealing conditions for finiteness and infinity of the Rado numbers.

Contribution

It provides exact formulas for the discrete and continuous 2-color Rado numbers for equations involving sums and linear relations, extending understanding of colorings in additive number theory.

Findings

Explicit formulas for R(m,c,2) depending on parity of m and c

Conditions under which the Rado number is infinite

Connection between discrete and continuous Rado numbers for linear equations

Abstract

For a positive integer $m$ and a real number $c$, let $R = R(m,c,2)$ denote the discrete 2-color Rado number for the equation $x_1 + x_2 + \dots + x_m + c = 2x_0$. In other words, $R$ is the smallest integer such that for any coloring of the integers ${1, 2, \dots, R}$, there exist numbers $x_1, x_2, \dots, x_m, x_0$, all with the same color, such that $x_1 + x_2 + \dots + x_m + c = 2x_0$. In this article we show that if $m \geq 2$ and $c > 0$, then $$ \begin{array}{cc} R(m,c,2) = \begin{cases} \infty & \text{for $m$ even, $c$ odd} \newline \big\lceil \frac{m}{2} \big\lceil \frac{m+c}{2} \big\rceil + \frac{c}{2} \big\rceil & \text{otherwise.} \end{cases} \end{array} $$ For real numbers $a$ and $c$, we look at the 2-color Rado number for the equation $x_1 + c = ax_0$. We show that if $a > 1$ and $c > 0$, then the 2-color continuous Rado number is $$ R_\mathbb{R}(1, c, a) = \begin{cases} \infty & \text{if $a=1$ } \newline \frac{c}{a-1} & \text{otherwise.} \end{cases} $$ From this, we will show that the discrete Rado number is $$ R(1, c, a) = \begin{cases} \frac{c}{a-1} & \text{if $ \left( a-1 \right) \mid c$} \newline \infty & \text{otherwise.} \end{cases} $$