We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$ -cliques, then $G$ admits a ${\it\mu}$ -measurable $d$ -coloring with respect to any Borel probability measure ${\it\mu}$ on $X$ , and a Baire measurable $d$ -coloring with respect to any compatible Polish topology on $X$ . The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$ -colorings of Cayley graphs of degree $d$ , except in two exceptional cases.

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

We construct an increasing ${\it\omega}$ -sequence $\langle \boldsymbol{a}_{n}\rangle$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each $\boldsymbol{a}_{n+1}$ is diagonally nonrecursive relative to $\boldsymbol{a}_{n}$ . It follows that the DNR principle of reverse mathematics does not imply the existence of Turing incomparable degrees.

For each positive $n$ , let $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ denote the identity obtained from the Adjan identity $(xy)(yx)(xy)(xy)(yx)\approx (xy)(yx)(yx)(xy)(yx)$ by substituting $(xy)\rightarrow (x_{1}x_{2}\ldots x_{n})$ and $(yx)\rightarrow (x_{n}\ldots x_{2}x_{1})$ . We show that every monoid which satisfies $\mathbf{u}_{n}\approx \boldsymbol{v}_{n}$ for each positive $n$ and generates a variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid $U_{2}(\mathbb{T})$ (respectively, $U_{2}(\overline{\mathbb{Z}})$ ) of two-by-two upper triangular tropical matrices over the tropical semiring $\mathbb{T}=\mathbb{R}\cup \{-\infty \}$ (respectively, $\overline{\mathbb{Z}}=\mathbb{Z}\cup \{-\infty \}$ ) is nonfinitely based.

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$ . Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$ -adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence having a unique model of cardinality $\aleph _{1}$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$ -Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$ -Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

We prove, in $\text{ZF}+\boldsymbol{{\it\Sigma}}_{2}^{1}$-determinacy, that, for any analytic equivalence relation $E$, the following three statements are equivalent: (1) $E$ does not have perfectly many classes, (2) $E$ satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class $[Y]_{E}$ we have that a real $X$ computes a member of the equivalence class if and only if ${\it\omega}_{1}^{X}\geqslant {\it\omega}_{1}^{[Y]}$. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over $ZF$.

Nešetřil and Ossona de Mendez introduced the notion of first-order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether, if (G i )i∈ℕ is a sequence of graphs with M being their first-order limit and v is a vertex of M, then there exists a sequence (v i )i∈ℕ of vertices such that the graphs G i rooted at v i converge to M rooted at v. We show that this holds for almost all vertices v of M, and we give an example showing that the statement need not hold for all vertices.

In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf ${\rm\Sigma}$-property’, Topology Appl.158(2) (2011), 204–214] by proving that under $CH$ a compact space $X$ is metrisable provided $X^{2}\setminus {\rm\Delta}$ can be covered by a family of compact sets $\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$ such that $K_{f}\subset K_{h}$ whenever $f\leq h$ coordinatewise.

We explain which Weierstrass ${\wp}$ -functions are locally definable from other ${\wp}$ -functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

The form of information presented can influence its utility for the conveying of knowledge by affecting an interpreter’s ability to reason with the information. There are distinct types of representational systems (for example, symbolic versus diagrammatic logics), various sub-systems (for example, propositional versus predicate logics), and even within a single representational system there may be different means of expressing the same piece of information content. Thus, to display information, choices must be made between its different representations, depending upon many factors such as: the context, the reasoning tasks to be considered, user preferences or desires (for example, for short symbolic sentences or minimal clutter within diagrammatic systems). The identification of all equivalent representations with the same information content is a sensible precursor to attempts to minimise a metric over this class. We posit that defining notions of semantic redundancy and identifying the syntactic properties that encapsulate redundancy can help in achieving the goal of completely identifying equivalences within a single notational system or across multiple systems, but that care must be taken when extending systems, since refinements of redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler-diagram-based notations. Such notations can be used to represent logical information and have applications including visualisation of database queries, social network visualisation, statistical data visualisation, and as the basis of more expressive diagrammatic logics such as constraint languages used in software specification and reasoning. The development of the new associated machinery and concepts required is important in its own right since it increases the growing body of knowledge on diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow for a much greater degree of flexibility of representation. We give syntactic properties that encapsulate semantic equivalence in both systems, whilst observing that the same semantic concept of redundancy is significantly more difficult to realise as syntactic properties in the extended system withprojections.

We show that a computable and conformal map of the unit disk onto a bounded domain $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ has a computable boundary extension if $D$ has a computable boundary connectivity function.

We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for $\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.

We obtain a formula for the reflexivity index of a finite lattice of sets and of various types of infinite lattices of sets.

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.