  This paper is an excerpt from the author's 1968 PhD dissertation [Yale
University, 1968] in which the (now) well-known result, commonly known as the
Folkman-Rado-Sanders theorem, is proved. The proof uses (finite) alternating
sums of integers and an 'iterated Ramsey theorem' in a way analogous to the
proof of Schur's theorem using differences of integers and Ramsey's theorem for
the coloring of the edges of a complete graph. The proof predates all others
except J. Folkman, who based his proof on van der Waerden's theorem. The paper
also contains the first published statement of the countable version of the
theorem, which came to be misattributed to Graham and Rothschild, but predated
their statement by three years.
