Related papers: The theory of hereditarily bounded sets

A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields…

We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all…

The usual definition of the set of constructible reals is $\Sigma ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be…

Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ integers from the set $\{1,...,m\}$ in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting…

The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG…

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of…

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan…

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of…

In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored.…

We prove that a hereditary class of graphs is $(\mathsf{tw}, \omega)$-bounded if and only if the induced minors of the graphs from the class form a $(\mathsf{tw}, \omega)$-bounded class.

We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…

We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…

Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange…

Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing…

The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two…