Related papers: A Ramsey-type phenomenon in two and three dimensio…
We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A…
We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…
A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $\rho$, every graph $G$ with sufficiently large $\rho(G)$ contains a `well-structured' induced subgraph $H$ with…
The measurable Riemann mapping theorem proved by Morrey and in some particular cases by Ahlfors, Lavrentiev and Vekua, says that any measurable almost complex structure on $\rd$ ($S^2$) with bounded dilatation is integrable: there is a…
Let $M$ be a locally symmetric irreducible closed manifold of dimension $\ge 3$. A result of Borel [Bo] combined with Mostow rigidity imply that there exists a finite group $G = G(M)$ such that any finite subgroup of $\text{Homeo}^+(M)$ is…
The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.
We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,\omega)$ of strong symplectic homeomorphisms, which…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
A Riemannian symmetric space is a Riemannian manifold in which it is possible to reflect all geodesics through a point by an isometry of the space. On such spaces, we introduce the notion of a distributional lattice, generalizing the notion…
The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic…
In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao.
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
We develop infinite-dimensional Ramsey theory for Fra\"iss\'e limits of finitely constrained free amalgamation classes in finite binary languages. We show that our approach is optimal and in particular, recovers the exact big Ramsey degrees…
We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it…
We introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev--Yaman's relative bounded cohomology for pairs. We obtain cohomological…
We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl Theorem, the author's Ramsey lift of bowtie-free graphs as well…
Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has…
We prove a reconstruction theorem for homeomorphism groups of open sets in metrizable locally convex topological vector spaces. We show that certain small subgroups of the full homeomorphism group obey the conditions of the above theorem.
We prove that for any homogeneous structure $\mathbf{K}$ in a language with finitely many relation symbols of arity at most two satisfying SDAP$^+$ (or LSDAP$^+$), there are spaces of subcopies of $\mathbf{K}$, forming subspaces of the…
The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, then any…