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This improves an earlier result of the first author for k = n = 2, namely Theorem II below, which dealt with B2 -sequences. This notion arose from a paper of Sidon [6]. A strictly increasing sequence A of natural numbers a,, a2 . . . . is called a B2-sequence, if for any two pairs (i, j) + (k, m) of natural numbers i <j, k < m there holds aj a i + ak a„„ in short, if A has no double-differences . (Of course it would be the same condition to presuppose that A has no double-sums : For any two pairs (i, j) + (k, m) with i < j, k < m there holds ai + a, + a k + a,,, .)

Congruent subsets of infinite sets of natural numbers.