Variations on van der Waerden’s and Ramsey’s Theorems

Most published proofs of van der Waerden’s theorem are carried out by induction on k and l. It wuld be of interest to find a direct inductive proof of Theorem V’. In this note, however, we shall show that Theorem V implies Theorem V’ and conversely. To this end, it is convenient to let V (k; l) denote the statement that if the set N of all positive integers is partitioned into k subsets, then at least one of these subsets contains an arithmetic progression of length l. Also, let V (m; l) denote the statement that if a1 < a2 < :: : are a sequence of positive integers such that a j+1 a j m, j = 1;2; : : : ; then the set fa1;a2; : : :g, contains an arithmetic progression of length l. Clearly Theorem V implies statement V (k; l) for every k and l, and Theorem V’ implies statement V (m; l) for every m and l. We show now that for every k and l, V (k; l) implies the existence of n(k; l). Indeed, suppose that the integer n= n(k; l) does not exist. Then for every n we have a sequence of length N on k symbols which represents a partition of f1;2; : : : ;ng into k subsets such that no subset contains a progression of length l. Let a1 be one of the k symbols which is the 1st symbol of infinitely many of these sequences. Let a2 be a symbol which is the 2nd symbol of infinitely many sequences beginning with a1. Let a3 be the 3rd symbol of infinitely many sequences starting with a1a2. In this way we construct an infinite sequence a1a3 on k symbols which represents a partition of N into k subsets, none of which contains