Modular Schur numbers

Jonathan Chappelon (1) , María Pastora Revuelta Marchena (2) , María Isabel Sanz Domínguez (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Departamento de Matemática Aplicada I

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 $3$.

Mots clés: Schur numbers; weak Schur numbers; modular Schur numbers; sum-free sets; weakly sum-free sets

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]

Jonathan Chappelon, María Pastora Revuelta Marchena, María Isabel Sanz Domínguez. Modular Schur numbers. The Electronic Journal of Combinatorics, 2013, 20 (2), pp.P61. ⟨10.37236/2374⟩. ⟨hal-00837647⟩