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 $x_1,x_2,\ldots,x_l,y$ satisfying the congruence $x_1+\ldots+x_l\equiv y\bmod{m}$. 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 $\{1,2,\ldots,n\}$ 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 $3$.
@article{10_37236_2374,
author = {Jonathan Chappelon and Mar{\'\i}a Pastora Revuelta Marchena and Mar{\'\i}a Isabel Sanz Dom{\'\i}nguez},
title = {Modular {Schur} numbers},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {2},
doi = {10.37236/2374},
zbl = {1295.11004},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2374/}
}
TY - JOUR
AU - Jonathan Chappelon
AU - María Pastora Revuelta Marchena
AU - María Isabel Sanz Domínguez
TI - Modular Schur numbers
JO - The electronic journal of combinatorics
PY - 2013
VL - 20
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/2374/
DO - 10.37236/2374
ID - 10_37236_2374
ER -
%0 Journal Article
%A Jonathan Chappelon
%A María Pastora Revuelta Marchena
%A María Isabel Sanz Domínguez
%T Modular Schur numbers
%J The electronic journal of combinatorics
%D 2013
%V 20
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/2374/
%R 10.37236/2374
%F 10_37236_2374
Jonathan Chappelon; María Pastora Revuelta Marchena; María Isabel Sanz Domínguez. Modular Schur numbers. The electronic journal of combinatorics, Tome 20 (2013) no. 2. doi: 10.37236/2374
