Andy Zucker
YOU?
Author Swipe
View article: Oscillating subalgebras of the atomless countable Boolean algebra
Oscillating subalgebras of the atomless countable Boolean algebra Open
We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.
View article: A survey on big Ramsey structures
A survey on big Ramsey structures Open
In recent years, there has been much progress in the field of structural Ramsey theory, in particular in the study of big Ramsey degrees. In all known examples of infinite structures with finite big Ramsey degrees, there is in fact a singl…
View article: Topological groups with tractable minimal dynamics
Topological groups with tractable minimal dynamics Open
A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class …
View article: Exact big Ramsey degrees for finitely constrained binary free amalgamation classes
Exact big Ramsey degrees for finitely constrained binary free amalgamation classes Open
We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization…
View article: POLISH SPACE PARTITION PRINCIPLES AND THE HALPERN–LÄUCHLI THEOREM
POLISH SPACE PARTITION PRINCIPLES AND THE HALPERN–LÄUCHLI THEOREM Open
The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles …
View article: Ultracoproducts and weak containment for flows of topological groups
Ultracoproducts and weak containment for flows of topological groups Open
We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. Fo…
View article: Minimal subdynamics and minimal flows without characteristic measures
Minimal subdynamics and minimal flows without characteristic measures Open
Given a countable group G and a G -flow X , a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$ -invariant. Frisch and Tamuz asked about the existence of a minimal G -flow, for any group G , which does…
View article: Ramsey theorem for trees with successor operation
Ramsey theorem for trees with successor operation Open
We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applica…
View article: Type-respecting amalgamation and big Ramsey degrees
Type-respecting amalgamation and big Ramsey degrees Open
We give an infinitary extension of the Nešetřil-Rödl theorem for category of relational structures with special type-respecting embeddings.
View article: Type-respecting amalgamation and big Ramsey degrees
Type-respecting amalgamation and big Ramsey degrees Open
We give an infinitary extension of the Nešetřil-Rödl theorem for category of relational structures with special type-respecting embeddings.
View article: Characterisation of the big Ramsey degrees of the generic partial order
Characterisation of the big Ramsey degrees of the generic partial order Open
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey …
View article: Infinite-dimensional Ramsey theory for binary free amalgamation classes
Infinite-dimensional Ramsey theory for binary free amalgamation classes Open
We develop infinite-dimensional Ramsey theory for Fraïssé 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 pr…
View article: Polish space partition principles and the Halpern-Läuchli theorem
Polish space partition principles and the Halpern-Läuchli theorem Open
The Halpern-Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles …
View article: From Ramsey degrees to Ramsey expansions via weak amalgamation
From Ramsey degrees to Ramsey expansions via weak amalgamation Open
Under no additional assumptions, in this paper we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgama…
View article: Minimal subdynamics and minimal flows without characteristic measures
Minimal subdynamics and minimal flows without characteristic measures Open
Given a countable group $G$ and a $G$-flow $X$, a measure $μ\in P(X)$ is called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal $G$-flow, for any group $G$, which does not adm…
View article: Topological dynamics beyond Polish groups
Topological dynamics beyond Polish groups Open
When G is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of G . We introduce a class of groups, the CAP groups, which provides a neat ge…
View article: Exact big Ramsey degrees via coding trees
Exact big Ramsey degrees via coding trees Open
We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization…
View article: Exact big Ramsey degrees for finitely constrained binary free amalgamation classes
Exact big Ramsey degrees for finitely constrained binary free amalgamation classes Open
We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization…
View article: Big Ramsey degrees of the generic partial order
Big Ramsey degrees of the generic partial order Open
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals).
View article: Bernoulli disjointness
Bernoulli disjointness Open
International audience
View article: A note on big Ramsey degrees
A note on big Ramsey degrees Open
Generalizing and simplifying recent work of Dobrinen, we show that if $\mathcal{L}$ is a finite binary relational language and $\mathcal{F}$ is a finite set of finite irreducible $\mathcal{L}$-structures, then the class $\mathcal{K} = \mat…
View article: On big Ramsey degrees for binary free amalgamation classes
On big Ramsey degrees for binary free amalgamation classes Open
Generalizing and simplifying recent work of Dobrinen, we show that if $\mathcal{L}$ is a finite binary relational language and $\mathcal{F}$ is a finite set of finite irreducible $\mathcal{L}$-structures, then the class $\mathcal{K} = \mat…
View article: Universal minimal flows of homeomorphism groups of high-dimensional\n manifolds are not metrizable
Universal minimal flows of homeomorphism groups of high-dimensional\n manifolds are not metrizable Open
Answering a question of Uspenskij, we prove that if $X$ is a closed manifold\nof dimension $2$ or higher or the Hilbert cube, then the universal minimal flow\nof $\\mathrm{Homeo}(X)$ is not metrizable. In dimension $3$ or higher, we also\n…
View article: A note on minimal models for pmp actions
A note on minimal models for pmp actions Open
Given a countable group , we say that a metrizable flow is model-universal if by considering the various invariant measures on , we can recover every free measure-preserving -system up to isomorphism. Weiss in[Dynamical systems and group …
View article: Fraïssé structures and a conjecture of Furstenberg
Fraïssé structures and a conjecture of Furstenberg Open
We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference betw…
View article: Topological dynamics of Polish group extensions
Topological dynamics of Polish group extensions Open
We consider a short exact sequence $1\to H\to G\to K\to 1$ of Polish groups and consider what can be deduced about the dynamics of $G$ given information about the dynamics of $H$ and $K$. We prove that if the respective universal minimal f…