Some Two Color, Four Variable Rado Numbers

There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},N\}$ admits a monochromatic solution to $x+y+kz=\mathrm{\ell }w$ for $k,\ell \in $\mathbb{Z}$^+$,where$N$dependson$k$and$\ell$\mathrm{.}Wedetermine$N$when$\ell-k \in {0,1,2,3,4,5}$,forall$k,\ell$forwhich${1/2}((\ell-k)^2-2)(\ell-k+1)\leq k \leq \ell-4$,aswellasforarbitrary$k$when$\ell=2$.