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 $x_1,x_2,\ldots,x_l,y$ 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 $3$.
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Submitted on : Monday, June 24, 2013 - 1:17:20 AM
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  • HAL Id : hal-00837647, version 1
  • ARXIV : 1306.5635

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

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