Some Two Color, Four Variable Rado Numbers

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 $\ell=2$.