Some two color, four variable Rado numbers

AbstractThere exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromatic solution to x+y+kz=ℓw for k,ℓ∈Z+, where N depends on k and ℓ. We determine N when ℓ−k∈{0,1,2,3,4,5}, for all k,ℓ for which 12((ℓ−k)2−2)(ℓ−k+1)⩽k⩽ℓ−4, as well as for arbitrary k when ℓ=2