Let ${\mathcal C} ^{\rm rt}$ be the class of rooted trees and $\langle {\mathcal C} ^{\rm rt}\rangle _{\dot{\cup }\Pi}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\mathbb X}= \dot{\bigcup} _{i<n}\prod _{j<m_i}{\mathbb X}_i^j$, where ${\mathbb X}_i^j$ are rooted trees. Defining ${\mathcal T}=\mathop{\rm Th} ({\mathbb X})$, ${\mathcal T} _i ^j=\mathop{\rm Th}({\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $\kappa = \prod _{i<n}\prod _{j<m_i}I({\mathcal T} _i^j)$, we have (a) Vaught's conjecture is true for ${\mathcal T}$: $I({\mathcal T})=\kappa $, if $\kappa\in \{ 1,\omega,{\mathfrak{c}}\}$, and, otherwise, $I({\mathcal T}) \in [3,\omega)$; (b) ${\mathbb Y} \equiv {\mathbb X}$ iff $\;{\mathbb Y} \cong \dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb Y} _i^j$, where ${\mathbb Y}_i^j\equiv {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (c) ${\mathbb E}\preccurlyeq {\mathbb X}$ iff $\;{\mathbb E} =\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb E}_i^j$, where ${\mathbb E}_i^j\preccurlyeq {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (d) ${\mathcal T}$ is atomic iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are atomic; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T}$, where ${\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T} _i^j$, for $i<n$ and $j<m_i$; (e) ${\mathcal T}$ is small iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are small; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}$, where ${\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}_i^j$, for $i<n$ and $j<m_i$.

We introduce a rank measure for first-order logic and prove a "rank-preserving'" version of Gaifman's theorem. Compared to earlier "rank-preserving locality theorems'" (in particular, [Grohe, Kreutzer, Siebertz, JACM 2017]), our theorem is not only much simpler, but also yields formulas in exactly the same normal form as Gaifman's original theorem. As an application of this theorem, we give a simplified proof of the main result of [Grohe, Kreutzer, Siebertz, JACM 2017] that first-order properties of nowhere-dense structures can be decided in almost linear time.

In 'On Denoting' Russell proposed the most influential theory of definite descriptions, expressions of the form 'the F'. Characteristic for Russell's approach is that definite descriptions are not treated as what they appear to be on the surface, i.e. as singular terms. Instead they are eliminated by a contextual definition. Russell formalises definite descriptions in the context of complete sentences of the form 'The F is G'. This requires scope markers to distinguish, e.g., internal from external negation. It was recognised by Burge, and Kalish and Montague, however, that the essential features of Russell's approach may be formalised while respecting the syntactic category to which definite descriptions appear to belong. An alternative, favoured by Neale, follows Russell in that complete sentences 'The F is G' are formalised by a binary quantifier. The undeniable importance of the theory of definite descriptions for logic, mathematics and philosophy demands that it be formalised to meet the standards of modern proof theory. This is the topic of the present paper. We systematise, compare and extend existing approaches. After presenting its essential features, we formalise Russell's theory of definite descriptions in sequent calculus. Three approaches will be considered. The first uses a binary quantifier, whereas the remaining two employ the term-forming iota operator. The first of these employs only the iota operator, the other employs in addition the lambda operator which does duty as a scope marker. All systems satisfy the standards for modern proof theory, in particular cut elimination. The appendix reformulates these systems in natural deduction, which is more convenient for practical purposes.

This paper proves normalisation theorems for intuitionist and classical positive free logic, without and with the iota operator for definite descriptions `the F'. Positive free logic also opens a number of options for rules for iota. In total, six different formalisations of theories of definite descriptions will be discussed, three proposed by Lambert, and three alternatives. The latter are motivated by considerations relating to proof-theoretic harmony between introduction and elimination rules. The philosophical importance of the various systems and results is indicated. The paper builds on Kürbis (2025), but is largely self-contained. The proofs for the present systems are easier than those for negative free logic.

Let $G$ be a locally finite multigraph that is bipartite and "unbalanced," meaning that it has a nontrivial bipartition $(A,B)$ with $°(x) > °(y)$ for all $x \in A$ and $y \in B$. We explore matchings in such graphs through the lens of descriptive set theory. In particular, we show that when $G$ is Borel and $\mu$ is a Borel probability measure on its vertex set, there is a Borel matching in $G$ that covers $\mu$-almost every vertex in $A$. This was previously known only under the assumption that $\mu$ is $G$-invariant, which we eliminate using a novel probabilistic approach. We also describe various extra conditions that imply the existence of a Borel matching covering every vertex in $A$. Along the way, we confirm a conjecture of the first and third named authors concerning the existence of Borel independent complete sections in Borel graphs of finite asymptotic separation index. In addition to their intrinsic interest, our results have applications to various other topics, such as edge-colorings, balanced orientations, and equidecomposition theory for group actions. For example, we show that the measurable edge-chromatic number of every Borel multigraph with finite maximum degree $\Delta$ is at most $\lfloor\frac{3\Delta}{2}\rfloor$, matching Shannon's optimal bound for finite multigraphs. Another example is that paradoxical Borel group actions with finite asymptotic separation index admit paradoxical decompositions with Borel pieces. This refines a result of Marks and Unger.

An $n$-partite graph is a graph such that every vertex has a color in $\{1,\ldots,n\}$ and every two vertices of the same color are not adjacent. We study the model comparisons of the theories of $n$-partite graph and $K_{\overline{m}}$-free $n$-partite graph, where $K_{\overline{m}}$ is a complete graph of a given size. The model companion of the theory of $n$-partite graph is simple and has IP. The model companion of the theory of $K_{\overline{m}}$-free $n$-partite graph has $\operatorname{TP}_2$, $\operatorname{SOP}_3$ and $\operatorname{NSOP}_4$ if $n > 2$. Forking independence coincides with dividing independence in this theory.

We continue to study ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances, together with the dc-embeddings, which we introduced in our earlier paper "Universal homogeneous two-sorted ultrametric spaces". The class of all finite two-sorted ultrametric spaces with dc-embeddings is Fraïssé whose limit we denote by $\mathbb{U}$. The main result of the article is that $\operatorname{Aut}(\mathbb{U})$ has a comeager conjugacy class. For that we show the cofinal amalgamation property of partial automorphisms and characterize amalgamation bases. In fact we develop a general strategy for showing cofinal amalgamation property for a broad class of categories. Furthermore, we show that there is no generic pair of automorphisms, we provide a detailed description of single orbits under dc-automorphisms, and we prove that any finite partial dc-automorphism, even in the presence of other orbits, can be extended to one that is closed or monotone.

Information is one of the most widely-discussed concepts of the current era. However, a great deal of insightful work notwithstanding, it is yet to be given wholly convincing logical or mathematical foundations. Without them, we lack adequate reasoning tools for understanding the complex ecosystems of systems upon which the society depends. We seek to rectify this by taking a first step towards developing an inferentialist semantic theory of information. There are three key interacting components. First, conceptual analysis: the metaphysics of information. Dretske expressed the key concepts of information in terms of intentionality, truth, and transmissibility. We replace truth with inferability, and trace the consequences of this replacement. Second, logic: proof-theoretic semantics (P-tS) provides a mathematical-logical realization of inferentialist reasoning. Using P-tS, we develop the first steps towards a mathematical-logical theory of an inferentialist primitive unit of information, the 'inferon'. This proof-theoretic approach counterpoints the model-theoretic view of information articulated in situation theory. Furthermore, we argue that it facilitates addressing all three components of van Benthem and Martinez's categorization of the understandings of information, as range, as correlation, and as code. Our focus is on information-as-correlation. Third, systems: the P-tS tools we develop provide the basis for a mathematical account of distributed systems modelling -- a key tool from informatics for understanding the organization of information processing systems. This yields a reasoning-based theory of information flow in models of distributed systems. Overall, we seek to give a conceptually rigorous mathematical-logical account of information and its role within informatics, grounded in inference and reasoning.

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then unless $D$ contains the complete symmetric directed graph on $d + 1$ vertices, $D$ admits a $μ$-measurable $d$-dicoloring with respect to any Borel probability measure $μ$ on $X$, and $D$ admits a $τ$-Baire-measurable $d$-dicoloring with respect to any Polish topology $τ$ compatible with the Borel structure on $X$. We also prove a definable version of Gallai's theorem on list dicolorings for directed graphs by showing that any Borel directed graph of bounded degree whose connected components are not Gallai trees is Borel degree-list-dicolorable.

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, $\text{NTP}_2$, $\text{NSOP}_1$, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between $\text{NSOP}_1$ and Kruckman and Ramsey's $\text{BTP}$ that applies to most known $\text{NSOP}_4$ examples in the literature.

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence.

Combining the methods of Brian and Stuart with the classical Dvoretzky theorem, we show that no infinite-dimensional Banach space contains a barrelled subspace of (algebraic) dimension $<\mbox{cov}(\mathcal{N})$, the covering number of the Lebesgue null ideal $\mathcal{N}$. Consequently, every infinite-dimensional normed barrelled space has dimension $\ge\mbox{cov}(\mathcal{N})$ and so it is consistent with \textsf{ZFC} that no normed barrelled space has dimension equal to the bounding number $\mathfrak{b}$.

We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c.\ groups, and show their equivalence. In the process, we obtain an algorithmic Stone-type duality between t.d.l.c.~groups and certain countable ordered groupoids given by the compact open cosets. We exploit the flexibility given by these different approaches to show that several natural groups, such as $\mathrm{Aut}(T_d)$ and $\mathrm{SL}_n(\mathbb Q_p)$, have computable presentations. We show that many construction leading from t.d.l.c.\ groups to new t.d.l.c.\ groups have algorithmic versions that stay within the class of computably presented t.d.l.c.\ groups. This leads to further examples, such as $\mathrm{PGL}_n(\mathbb Q_p)$. We study whether objects associated with computably t.d.l.c.\ groups are computable: the modular function, the scale function, and Cayley-Abels graphs in the compactly generated case. We give a criterion when computable presentations of t.d.l.c.~groups are unique up to computable isomorphism, and apply it to $\mathbb Q_p$ as an additive group, and the semidirect product $\mathbb Z\ltimes \mathbb Q_p$. We give (joint with Willis) an example of a computably t.d.l.c. group with noncomputable scale function.

We build models using an indiscernible model sub-structures of ${\kappa} \ge {\lambda}$ and related more complicated structures. We use this to build various Boolean algebras.