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 $x_1,x_2,...,x_l,y$ satisfying the congruence $x_1+\dot{.}.+x_l\equiv y\mathrm{mod}m$. 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 $\{1,2,\dot{.}.,n\}$ 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.

arXiv comment25 pages, 3 tables