Off-Diagonal Continuous Rado Numbers $x_1 + x_2 + \dots + x_k = x_0$

TL;DR

The paper determines the exact continuous off-diagonal Rado number for two-colourings: the smallest real $S$ such that any red/blue coloring of the interval $[1,S]$ contains a red solution to $x_1+\cdots+x_k=x_0$ or a blue solution to $x_1+\cdots+x_l=x_0$ with $2\le k\le l$, proving $S_\,mathrm{R}(k,l)=kl+k-1$. It furnishes a lower bound via an explicit coloring and an upper bound by treating the $k=2$ case directly and adapting the Robertson–Schaal framework for $k\ge3$ with two fixes (Lemmas 3 and 4) to handle the blue-1 scenario, culminating in Theorem 2 and a gamma-parameterized extension (Theorem 3). The work unifies the $k\ge3$ case with the discrete RS results and provides a streamlined, continuous analogue for the $k=2$ case, with potential extensions to more colors and other equation families. This introduces a robust continuous counterpart to classical Ramsey–Rado theory, offering exact thresholds and methods for off-diagonal systems in real intervals.

Abstract

In 2001, Robertson and Schaal found the 2-color off-diagonal generalized Schur numbers: for two positive integers $k$ and $l$, they determined the smallest positive integer $S = S(k, l)$ such that for any coloring of the integers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$. We extend this result to find the continuous version: for two positive integers $k$ and $l$, we find the smallest real number $S = S_\mathbb{R} (k, l)$ such that for any coloring of the real numbers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$.