We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$ -free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

We build and study dynamic versions of epistemic logic. We study languages parameterized by an action signature that allows one to express epistemic actions such as (truthful) public announcements, completely private announcements to groups of agents, and more. The language $\mathcal{L}$(Σ) is modeled on dynamic logic. Its sentence-building operations include modalities for the execution of programs, and for knowledge and common knowledge. Its program-building operations include action execution, composition, repetition, and choice. We consider two fragments of $\mathcal{L}(\mathbf{\Sigma })$. In ${\mathcal{L}}_1(\mathbf{\Sigma })$, we drop action repetition; in ${\mathcal{L}}_0(\mathbf{\Sigma })$, we also drop common knowledge. We present the syntax and semantics of these languages and sound proof systems for the validities in them. We prove the strong completeness of a logical system for ${\mathcal{L}}_0(\mathbf{\Sigma })$ and the weak completeness of one for ${\mathcal{L}}_1(\mathbf{\Sigma })$. We show the finite model property and, hence, decidability of ${\mathcal{L}}_1(\mathbf{\Sigma })$. We translate ${\mathcal{L}}_1(\mathbf{\Sigma })$ into PDL, obtaining a second proof of decidability. We prove results on expressive power, comparing ${\mathcal{L}}_1(\mathbf{\Sigma })$ with modal logic together with transitive closure operators. We prove that a logical language with operators for private announcements is more expressive than one for public announcements.

We investigate the structure of Gδ ideals of compact sets. We define a class of Gδ ideals of compact sets that, on the one hand, avoids certain phenomena present among general Gδ ideals of compact sets and, on the other hand, includes all naturally occurring Gδ ideals of compact sets. We prove structural theorems for ideals in this class, and we describe how this class is placed among all Gδ ideals. In particular, we establish a result representing ideals in this class via the meager ideal. This result is analogous to Choquet's theorem representing alternating capacities of order ∞ via Borel probability measures. Methods coming from the structure theory of Banach spaces are used in constructing important to us examples of Gδ ideals outside of our class.

We show that, for a generic measure preserving transformation T, the closed group generated by T is not isomorphic to the topological group \(L^0(\lambda , {{\mathbb {T}}})\) of all Lebesgue measurable functions from [0, 1] to \({\mathbb {T}}\) (taken with pointwise multiplication and the topology of convergence in measure). This result answers a question of Glasner and Weiss. The main step in the proof consists of showing that Koopman representations of ergodic boolean actions of \(L^0(\lambda , {\mathbb T})\) possess a non-trivial spectral property not shared by all unitary representations of \(L^0(\lambda , {{\mathbb {T}}})\). The main tool underlying our arguments is a theorem on the form of unitary representations of \(L^0(\lambda , {{\mathbb {T}}})\) from our earlier work.

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem.