Related papers: On Schreier-type Sets, Partitions, and Composition…

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+\beta)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results…

We study the model theoretic structure $(\Z,+,P_r)$ where $r>1$ is an irrational number and the elements of $P_r$ are of the form $\floor{nr}$ for some $n\in\Z\setminus\{0\}$. We axiomatize of this structure and prove a quantifier…

For a set of nonnegative integers $S$ let $R_{S}(n)$ denote the number of unordered representations of the integer $n$ as the sum of two different terms from $S$. In this paper we focus on partitions of the natural numbers into two sets…

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. Erd\H{o}s…

Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $\nu$ non-empty subsets $S_1,\dots, S_{\nu}$ is called perfect, if all $\nu$ values…

We give an asymptotic estimate for the number of partitions of a set of $n$ elements, whose block sizes avoid a given set $\mathcal{S}$ of natural numbers. As an application, we derive an estimate for the number of partitions of a set with…

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr…

A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that…

We say that a set $S$ is $\Delta^0_{(n)}(X)$ if membership of $n$ in $S$ is a $\Delta^0_{n}(X)$ question, uniformly in $n$. A set $X$ is low for $\Delta$-Feiner if every set $S$ that is $\Delta^0_{(n)}(X)$ is also…

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we…

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…

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…

If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We…

Consider the set $\{1,2,\ldots,3n\}$. We are interested in the number of partitions of this set into subsets of three elements each, where the sum of two of them equals the third. We give some criteria such a partition has to fulfill, which…

We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1…

In this paper we present a new formula for the number of unrestricted partitions of $n$. We do this by introducing a correspondence between the number of unrestrited partitions of $n$ and the number of non-negative solutions of systems of…

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper…

Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is…

A partition of a positive integer $n$ is defined as a non-increasing sequence $P = [y_0, y_1, ..., y_m]$ of positive integers which sum to $n$, where the $y_i$ are called the $parts$ of the partition. A Young diagram is a visual…

Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences…