Related papers: Sum-free sets in abelian groups

We study progression-free sets in the abelian groups $G=(\mathbb{Z}_m^n,+)$. Let $r_k(\mathbb{Z}_m^n)$ denote the maximal size of a set $S \subset \mathbb{Z}_m^n$ that does not contain a proper arithmetic progression of length $k$. We give…

A subset $A$ of a group $G$ is called $(k, l)$-{\it sumset}, if $A= kB-lB$ for some $B\subseteq G$, where $kB-lB={x_1+...+x_k-x_{k+1}-...-x_{k+l} : x_1,..., x_{k+l}\in B}.$ Upper and lower bounds for the number $(k, l)$-sumsets in groups of…

In this note we provide a negative answer to the question: ``Is it true that for every positive rational number $r$ there exists a finite abelian group $G$ such that $|\mathrm{Aut}(G)|/|G| = r$?". We show that if $r = a/b$ is a rational…

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

Let G be a finite Abelian group and A be a subset G\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This…

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show,…

We show that for any set A in a finite Abelian group G that has at least c |A|^3 solutions to a_1 + a_2 = a_3 + a_4, where a_i belong A there exist sets A' in A and L in G, |L| \ll c^{-1} log |A| such that A' is contained in Span of L and…

A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are…

Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of cardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ has size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost fully…

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of…

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,…

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…

In this short note we show that if lambda>aleph_1 is regular and lambda is not the successor of a singular cardinal of cofinality aleph_0, and G is a lambda-free abelian group of size lambda, then there is a free group G' subseteq G of size…

Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < { and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every…

Let $G$ be an additive finite abelian group of order $n$, and let $S$ be a sequence of $n+k$ elements in $G$, where $k\geq 1$. Suppose that $S$ contains $t$ distinct elements. Let $\sum_n(S)$ denote the set that consists of all elements in…

Let $G$ be a nonabelian group, $A\subseteq G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting…

There has been much work on the following question: given n how large can a subset of {1,...,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch…

For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of…

In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is examined when $G$ is a finite nilpotent group, finite abelian group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive the…

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when…