Rado numbers for \(a(x+y)bz\)
A well-known theorem of Schur guarantees for any number of colors \(k\) the existence of a minimal integer \(S(k)\) such that any \(k\)-coloring of the numbers \(1,2,\dots, S(k)\) contains a monochromatic solution of the equation \(x+y= z\). The authors consider the more general case \(a(x+ y)= bz\). They denote by \(Ra_k(a,b)\) a generalization of the number \(S(k)\) \((Ra_k(a, a)= S(k))\). The main results of this paper are formulas describing the values of \(Ra_2(a, b)\) for \(b= 2\) and \(b= a+1\), namely: \(Ra_2(a, 2)= a(a^2+ 1)/2\) and \(Ra_2(a, a+1)= a(a+ 1)\) for \(a\geq 3\), and a computer-aided calculation of \(Ra_k(a, b)\) for some \(a,b\leq 23\), \(k= 2\) and \(a,b\leq 5\), \(k= 3\). Moreover, they consider a further generalization of the equation \(a(x+y)= bz\) to a system of \(m\) equations \(a(x_n+ x_{n+1})= bx_{n+2}\), \(n= 1,\dots,m\). Furthermore, the authors define numbers \(Ra_k(m; a,b)\) such that \(Ra_k(1; a,b)= Ra_k(a, b)\).
scientific article; zbMATH DE number 1093111