Determination of the two-color Rado number for

For positive integers a 1 , a 2 , … , a m , we determine the least positive integer R ( a 1 , … , a m ) such that for every 2-coloring of the set [ 1 , n ] = { 1 , … , n } with n ⩾ R ( a 1 , … , a m ) there exists a monochromatic solution to the equation a 1 x 1 + ⋯ + a m x m = x 0 with x 0 , … , x m ∈ [ 1 , n ] . The precise value of R ( a 1 , … , a m ) is shown to be a v 2 + v − a , where a = min { a 1 , … , a m } and v = ∑ i = 1 m a i . This confirms a conjecture of B. Hopkins and D. Schaal.