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 $1/2((\ell -k{)}^2-2)(\ell -k+1)\le k\le \ell -4$, as well as for arbitrary $k$ when $\ell =2$.

13 pages