Related papers: Roth-type theorems in additive combinatroics

De Bruijn and Erd\H{o}s proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on…

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The…

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B.…

A major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemer\'edi's theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs…

In 1940 Paul Erd\H{o}s made a conjecture about the distribution of reduced residues. Here we study the distribution of $k$-tuple of reduced residues.

Combinatorial methods (or methods of elementary transformations) came to group theory from low-dimensional topology in the beginning of the century. Soon after that, combinatorial group theory became an independent area with its own…

We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.

We observe that a recent result by Gardiner and Sj\"odin, solving a problem of Kr\'{a}l on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which…

We prove a Roth type theorem for polynomial corners in the finite field setting. Let $\phi_1$ and $\phi_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with…

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erd\H{o}s-Rado Theorem--into the notion of a combinatorial Erd\H{o}s-Rado class. These classes are closely related to Erd\H{o}s-Rado classes, which are those from…

The famous van der Waerden theorem states that if partition N into finitely many cells then one of them will contain arbitrary length arithmetic progressions. It has a polynomial version also. In this article we will prove the near 0…

This is an English translation of Reidemeister's book "Einf\"uhrung in die kombinatorische Topologie" from 1932, the first monograph on combinatorial group theory and topology, with some added comments by the translator and Warren Dicks.

We consider a combinatorial problem occurring naturally in a group theoretical setting and provide a constructive solution in a special case. More precisely, in 1999 the author established a logarithmic bound for the derived length of the…

The Erd\H{o}s-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations…

Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…

In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof…

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories…

In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any $n+3$ general position points in…

We consider the immediate consequence of an arguable addition to the standard Deduction Theorems of first order theories.