Related papers: Beyond sum-free sets in the natural numbers
A set A of integers is said to be sum-free if there are no solutions to the equation x + y = z with x,y and z all in A. Answering a question of Cameron and Erdos, we show that the number of sum-free subsets of {1,...,N} is O(2^(N/2)).
A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least…
A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove…
We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.
$S \subseteq \mathbb{Z}_{2n}$ is said to be sum-free if $S$ has no solution to the equation $a+b=c$. The sum-free process on $\mathbb{Z}_{2n}$ starts with $S:=\emptyset$, and iteratively inserts elements of $\mathbb{Z}_{2n}$, where each…
Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal…
A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements.…
In this paper we rectify two previous results found in the literature. Our work leads to a new upper bound for the largest sum-free subset of $[1,n]$ with lowest value in $\left [\frac{n}{3},\frac{n}{2}\right ]$, and the identification of…
Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The…
For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically,…
Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…
A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not-necessarily-distinct) elements of $A$ never equals the sum of $l$ (not-necessarily-distinct) elements of $A$. We find an explicit formula for the…
Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…
In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…
The additive closedness in the subset of an additive group is termed as r-value. The nature of closedness in different subsets of fixed size is observed as a spectrum of r-values. We enumerate r-values of subsets in finite fields of…
Cameron introduced a bijection between the set of sum-free sets and the set of all zero-one sequences. In this paper, we study the sum-free sets of natural numbers corresponding to certain zero-one sequences which contain the Cantor-like…
In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem…
A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding…
For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…
Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally…