Related papers: On maximal sum-free sets in abelian groups

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in…

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we…

It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands,…

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…

We show that a finite zero-sum-free sequence $\alpha$ over an abelian group has at least $c|\alpha|^{4/3}$ distinct subsequence sums, unless $\alpha$ is "controlled" by a small number of its terms; here $|\alpha|$ denotes the number of…

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…

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2,…

We show that a zero-sum-free sequence of length $n$ over an abelian group spans at least $2n$ distinct subsequence sums, unless it possesses a rigid, easily-described structure.

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…

We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull…

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture,…

Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally…

The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the M\"obius function is given for the general abelian group, while in many special cases, including the case of…

The number of maximal abelian subgroups of a finite p-group is shown to be congruent to 1 modulo p.

A set $A\subset \mathbb{F}_p^n$ is sum-free if $A+A$ does not intersect $A$. If $p\equiv 2 \mod 3$, the maximal size of a sum-free in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset…

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson…

A sequence in the additive group ${\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ and sum zero. The article characterizes the $n$-zero-free sequences in ${\mathbb Z}_n$ of…

Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a…

We study the maximal cross number $\mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $\mathsf{k}(G)$ of a zero-sum free sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the first part of…