Related papers: Algebra, selections, and additive Ramsey theory
The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model…
We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…
Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In…
We show that: 1. Rothberger bounded subgroups of sigma-compact groups are characterized by Ramseyan partition relations. 2. For each uncountable cardinal $\kappa$ there is a ${\sf T}_0$ topological group of cardinality $\kappa$ such that…
It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective…
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability…
For $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B…
Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…
In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by…
For any subset $Z \subseteq \mathbb{Q}$, consider the set $S_Z$ of subfields $L\subseteq \overline{\mathbb{Q}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in $L$ such that $C \cap \mathbb{Q}=Z$. Placing…
Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses M\"obius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory.…
As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a…
In this paper we settle long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of…
Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid. As our main tool, we prove for any infinite graph $G$ with vertex sets $A$…
The Ramsey Choice principle for families of $n$-element sets, denoted $\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$. We investigate…
The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and…
A linear system on a smooth complex algebraic surface gives rise to a family of smooth curves in the surface. Such a family has a topological monodromy representation valued in the mapping class group of a fiber. Extending arguments of…
We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…
We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive…
Finite games in normal form and their mixed extensions are a corner stone of noncooperative game theory. Often generic finite games and their mixed extensions are considered. But the properties which one expects in generic games and the…