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

TL;DR

This paper extends the concept of off-diagonal Schur numbers from integers to real numbers, determining the minimal continuous interval ensuring monochromatic solutions for specific linear equations under any two-coloring.

Contribution

The authors generalize discrete off-diagonal Schur numbers to the continuous setting, deriving the minimal real interval guaranteeing monochromatic solutions for the equations.

Findings

Derived the value of $S_ ext{R}(k, l)$ for continuous real numbers.

Extended discrete combinatorial results to the real number continuum.

Established conditions for monochromatic solutions in the continuous case.

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 $x_1 + x_2 + \dots + x_l = x_0$.