Related papers: Model theory and combinatorics of banned sequences

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in $o$-minimal expansions of fields. Using it, we…

Clozel, Harris and Taylor proposed conjectural generalizations of the classical Ihara's lemma for $\mathrm{GL}_2$, to higher dimensional similitude groups. We prove these conjectures in the so called limit case, which after base change is…

We define a new class of infinitary logics $\mathscr L^1_{\kappa,\alpha}$ generalizing Shelah's logic $\mathbb L^1_\kappa$ defined in \cite{MR2869022}. If $\kappa=\beth_\kappa$ and $\alpha <\kappa$ is infinite then our logic coincides with…

The cosection lemma proved by J. Li and Y.H. Kiem said the intrinsic normal cone lies inside the kernel of any cosection of the obstruction sheaf when the moduli has a perfect obstruction theory. With a definition of higher tangent vectors…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

In this note we develop and clarify some of the basic combinatorial properties of the new notion of $n$-dependence (for $1\leq n < \omega$) recently introduced by Shelah. In the same way as dependence of a theory means its inability to…

In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…

In spite of its many successes, the Standard Model makes many empirical assumptions in the Higgs and fermion sectors for which a deeper theoretical basis is sought. Starting from the usual gauge symmetry $u(1) \times su(2) \times su(3)$…

We prove limit theorems for sums of randomly chosen random variables conditioned on the summands. We consider several versions of the corner growth setting, including specific cases of dependence amongst the summands and summands with heavy…

An order-theoretic generalization of Seymour relations describing the connection between the set-theoretic blocker, deletion, and contraction maps on clutters, is presented.

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of…

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…

This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…

We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present…

Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.

This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.

We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows us to deduce complete theories of given models. The first instance of such rules was the $\omega$-rule introduced by Hilbert,…

We set up some foundations of generalised scheme theory related to new incompressible symmetric tensor categories. This is analogous to the relation between super schemes and the category of super vector spaces.

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate…

We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma, in conjunction with a result of Buzzard's, to give a proof of (a generalization) of Coleman's theorem which states that overconvergent…