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We continue algebraization of the set of ultrafilters on a metric spaces initiated in [6]. In particular, we define and study metric counterparts of prime, strongly prime and right cancellable ultrafilters from the Stone-$\check{C}$ech…
Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from…
We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…
N. Hindman and D. Strauss had shown that, for discrete semigroups, the cartesian product of two central sets are central. They also proved that the product of J- sets and C-sets are also J-set and C-set and characterized when the infnite…
Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size \kappa satisfying a large fragment of ZFC. We introduce new large cardinal axioms…
A new method of analysing positive bistochastic maps on the algebra of complex matrices $M_{3}$ has been proposed. By identifying the set of such maps with a convex set of linear operators on $\mathbb{R}^{8}$, one can employ techniques from…
A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By…
We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…
Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the…
$IP$ sets play fundamental role in arithmetic Ramsey theory. A set is called an additive $IP$ set if it is of the form $FS\left(\langle x_{n}\rangle_{n\in \mathbb{N}}\right)=\left\{ \sum_{t\in H}x_{t}:H\right.$ is a nonempty finite subset…
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is…
Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has…
Ramsey theory and forcing have a symbiotic relationship. At the RIMS Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave three tutorials on Ramsey theory in forcing. The first two tutorials concentrated on…
A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in…