| For every integer c and every positive integer m ≥ 3, let n = R(m, c) be the least integer, provided that it exists, such that for every coloring D:1,2&ldots;,n→ 0,1, there exists integers x1, x2,...,xm (not necessarily distinct) such that Dx1= Dx2=&cdots; =Dxm and x1+x2+&cdots;+xm-1+c= m-1xm. If such an integer does not exist, then let R( m, c) = infinity. The main result is that for every odd integer m ≥ 3 and every positive integer c Rm,c= 2&ceill0;cm-1&ceilr0;+1 ifcis eveninfinityif cisodd. . |
