Some Two Color, Four Variable Rado Numbers

TL;DR

This paper investigates the minimal number N such that any 2-coloring of the set {1,...,N} guarantees a monochromatic solution to a specific linear equation involving four variables, extending known results for certain parameter ranges.

Contribution

The paper determines exact Rado numbers for a class of linear equations with two colors, covering specific parameter ranges and general cases for certain variables.

Findings

Calculated Rado numbers for , , , ,  cases.

Extended known bounds for monochromatic solutions in 2-colorings.

Provided explicit formulas for minimal N based on parameters k and .

Abstract

There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$. We determine $N$ when $\ell-k \in \{0,1,2,3,4,5\}$, for all $k,\ell$ for which ${1/2}((\ell-k)^2-2)(\ell-k+1)\leq k \leq \ell-4$, as well as for arbitrary $k$ when $\ell=2$.