There exists a minimum integer N such that any 2-coloring of
{1,2,...,N} admits a monochromatic solution to x+y+kz=βw for k,ββZ+, where N depends on k and β. We determine N when
ββkβ{0,1,2,3,4,5}, for all k,β for which
1/2((ββk)2β2)(ββk+1)β€kβ€ββ4, as well as for arbitrary k
when β=2.Comment: 13 page