Modular schur numbers

Abstract

For any positive integers l and m, a set of integers is said to be (weakly) /-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 I, 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.