scholarly journals Modular Schur Numbers

10.37236/2374 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Chappelon ◽  
María Pastora Revuelta Marchena ◽  
María Isabel Sanz Domínguez
Keyword(s):  
Positive Integers ◽  
Free Sets

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 $3$.

On the number of sum-free sets in a segment of positive integers

2002 ◽  
Vol 12 (4) ◽  
Author(s):  
K. G. Omelyanov ◽  
A. A. Sapozhenko
Keyword(s):  
Basic Research ◽  
Basic Research Grant ◽  
Positive Integers ◽  
Free Sets ◽  
Research Grant

AbstractA set A of integers is called sum-free if a + b ∉ A for any a, b ∈ A. For an arbitrary Ɛ > 0, let ssThis research was supported by the Russian Foundation for Basic Research, grant 01-01-00266.

scholarly journals A Note on the Number of $(k,l)$-Sum-Free Sets

10.37236/1508 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Tomasz Schoen
Keyword(s):  
Natural Number ◽  
Positive Integers ◽  
Free Sets

A set $A\subseteq {\bf N}$ is $(k,\ell)$-sum-free, for $k,\ell\in {\bf N}$, $k>\ell$, if it contains no solutions to the equation $x_1+\dots+x_k=y_1+\dots+y_{\ell}$. Let $\rho=\rho (k-\ell)$ be the smallest natural number not dividing $k-\ell$, and let $r=r_n$, $0\le r < \rho$, be such that $r\equiv n \pmod {\rho }$. The main result of this note says that if $(k-\ell)/\ell$ is small in terms of $\rho$, then the number of $(k,\ell)$-sum-free subsets of $[1,n]$ is equal to $(\varphi(\rho)+\varphi_r(\rho)+o(1)) 2^{\lfloor n/\rho \rfloor}$, where $\varphi_r(x)$ denotes the number of positive integers $m\le r$ relatively prime to $x$ and $\varphi(x)=\varphi_x(x)$.

Almost Odd Random Sum-Free Sets

1998 ◽  
Vol 7 (1) ◽  
pp. 27-32 ◽  
Author(s):  
NEIL J. CALKIN ◽  
P. J. CAMERON
Keyword(s):  
Random Sum ◽  
Positive Probability ◽  
Finite Sum ◽  
Free Set ◽  
Positive Integers ◽  
Free Sets

We show that if S1 is a strongly complete sum-free set of positive integers, and if S0 is a finite sum-free set, then, with positive probability, a random sum-free set U contains S0 and is contained in S0∪S1. As a corollary we show that, with positive probability, 2 is the only even element of a random sum-free set.

scholarly journals Maximal (k, l)-free sets in ℤ/pℤ are arithmetic progressions

2002 ◽  
Vol 65 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Alain Plagne
Keyword(s):  
Arithmetic Progressions ◽  
Recent Issue ◽  
Complete Determination ◽  
Free Set ◽  
Positive Integers ◽  
Free Sets

Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.

scholarly journals Maximum $k$-Sum $\mathbf{n}$-Free Sets of the 2-Dimensional Integer Lattice

10.37236/8895 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Ilkyoo Choi ◽  
Ringi Kim ◽  
Boram Park
Keyword(s):  
Positive Integer ◽  
Lattice Point ◽  
Higher Dimensions ◽  
Maximum Density ◽  
Integer Lattice ◽  
Free Set ◽  
Integer Lattice Point ◽  
Positive Integers ◽  
Free Sets

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements $\mathbf{a}_1, \ldots, \mathbf{a}_k$ in $S$ satisfying $\mathbf{a}_1+\cdots+\mathbf{a}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a $k$-sum $\mathbf{b}$-free set of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions. 

scholarly journals The Structure of Maximum Subsets of $\{1,\ldots,n\}$ with No Solutions to $a+b = kc$

10.37236/1916 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Andreas Baltz ◽  
Peter Hegarty ◽  
Jonas Knape ◽  
Urban Larsson ◽  
Tomasz Schoen
Keyword(s):  
Positive Integer ◽  
Precise Characterization ◽  
Positive Integers ◽  
Free Sets

If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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