Related papers: Random sum-free subsets of Abelian groups

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group.…

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting.…

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…

In this paper we highlight a few open problems concerning maximal sum-free sets in abelian groups. In addition, for most even order abelian groups $G$ we asymptotically determine the number of maximal distinct sum-free subsets in $G$. Our…

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise the largest possible sum-free subsets of G…

In this paper we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind, and is hence not overly technical. We consider answers to questions like: how…

A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian…

Given an Abelian groups $G$, denote $\mu(G)$ the size of its largest sum-free subset and $f_{\max}(G)$ the number of maximal sum-free sets in $G$. Confirming a prediction by Liu and Sharifzadeh, we prove that all even-order $G\ne…

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…

In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>\frac{2}{7}n. In this note, we show that the argument…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

It is well-known that for a prime $p\equiv 2\pmod 3$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $\mathbb Z_p^n$ is $\frac13\,(p+1)p^{n-1}$. We establish a matching stability result…

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there…

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…

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$…

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erd\H{o}s. In…

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect…

Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq G$ be a random set of size $|G|^\alpha$. We let $$ \mu(A) = \max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|. $$ The issue is to determine…

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a…

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…