Related papers: On pairs of definable orthogonal families

A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit…

We study maximal orthogonal families of Borel probability measures on $2^\omega$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a…

We show that for any $k\in\omega$, the structure $(H_k,\in)$ of sets that are hereditarily of size at most $k$ is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds…

We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.

We prove that any definable family of subsets of a definable infinite set $A$ in an o-minimal structure has cardinality at most $|A|$. We derive some consequences in terms of counting definable types and existence of definable topological…

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(\omega, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can…

In this paper we consider nonmeasurablity with respect to sigma-ideals defined be trees. First classical example of such ideal is Marczewski ideal s_0. We will consider also ideal l_0 defined by Laver trees and m_0 defined by Miller trees.…

We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our…

We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof…

In the additive topological group $(\mathbb{R},+)$ of real numbers, we construct families of sets for which elements are not measurable in the Lebesgue sense. The constructed families have algebraic structures of being semigroups (i.e.,…

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make…

In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or $ln-exp$…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

The notion of $\alpha$-large families of finite subsets of an infinite set is defined for every countable ordinal number $\alpha$, extending the known notion of large families. The definition of the $\alpha$-large families is based on the…

The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of…

We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain "algebraic patterns" in every subset in the family. By…

We study families of subsets of $\omega$ which are independent with respect to the asymptotic density $\mathsf{d}$. We show, for instance, that there exists a maximal $\mathsf{d}$-independent family $\mathcal{A}$ such that…

For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real…

A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power…

We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general…