Rado numbers for a(x+y)=bz

摘要

In the case of existence the smallest number N = Ra-k is called a Rado number if it is guaranteed that ant k-coloring of the numbers 1, 2,..., N contains a monochromatic solution of a given system of linear equations. We will determine Ra-k(a, b) for the equation a(x + y) = bz if b = 2 and b = a + 1. Also, the case of monochromatic sequences {x(n)} generated by a(x(n) + x(n+1)) = bx(n+2) is discussed. (C) 1997 Academic Press.