Determination of the two-color Rado number for a1x1+⋯+amxm=x0

For positive integers a1,a2,…,am, we determine the least positive integer R(a1,…,am) such that for every 2-coloring of the set [1,n]={1,…,n} with n⩾R(a1,…,am) there exists a monochromatic solution to the equation a1x1+⋯+amxm=x0 with x0,…,xm∈[1,n]. The precise value of R(a1,…,am) is shown to be av2+v−a, where a=min{a1,…,am} and . This confirms a conjecture of B. Hopkins and D. Schaal.