Modular Schur numbers

Abstract

For any positive integers l and m, a set of integers is said to be (weakly) l-sum free modulo m if it contains no (pairwise distinct) elements x1, x2, . . . , xl , y satisfying the congruence x1 + . . . + xl ≡ y mod m. It is proved that, for any positive integers k and l, there exists a largest integer n for which the set of the first n positive integers {1, 2, . . . , n} admits a partition into k (weakly) l-sum-free sets modulo m. This number is called the generalized (weak) Schur number modulo m, associated with k and l. In this paper, for all positive integers k and l, the exact value of these modular Schur numbers are determined for m = 1, 2 and 3.