On Rado numbers for ∑i=1m−1aixi=xm

For all integers m⩾3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to a1x1+a2x2+⋯+am−1xm−1=xm. Let t=min{a1,a2,…,am−1} and b=a1+a2+⋯+am−1−t. In this paper it is shown that whenever t=2, R(a1,a2,…,am−1)=2b2+9b+8. It is also shown that for all values of t, R(a1,a2,…,am−1)⩾tb2+(2t2+1)b+t3.