Related papers: On the Effectiveness of Partition Regularity over …
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur's celebrated partition identity (1926). Andrews' two generalisations of Schur's theorem went on to become two of the most…
The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher's theorem: for all $m \geq 2$ and…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of…
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
A finite or infinite matrix A with rational entries is called partition regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey Theory may naturally…
Graphs on surfaces is an active topic of pure mathematics belonging to graph theory. It has also been applied to physics and relates discrete and continuous mathematics. In this paper we present a formal mathematical description of the…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and…
Categories of partitions are combinatorial structures arising from the representation theory of certain compact quantum groups and are linked to classical diagram algebras such as the Temperley-Lieb algebra. In this paper, we present…
Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual…
We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure…
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the author's recent method of trees with coding nodes are…
In recent years, there has been much progress in the field of structural Ramsey theory, in particular in the study of big Ramsey degrees. In all known examples of infinite structures with finite big Ramsey degrees, there is in fact a single…
We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.
In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the…
We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ when $1 < c < c^\dag(s)$, where $s \geqslant 3$ is the number of variables. Here $c^\dag(3) = 12/11$ and $c^\dag(4) = 7/6$,…
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…