A variant of the Hales-Jewett Theorem

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that {b(a+id)^j:i,j \in\nhat k}\subset B. In particular one cell of each finite partition of the positive integers contains such configurations. We prove a Hales-Jewett type extension of this partition theorem

Partition regularity and multiplicatively syndetic sets

We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green-Tao and Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x, d, x + d, x + 2d\}$.Comment: 29 pages. v3. Referee comments incorporated, accepted for publication in Acta Arithmetic

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry

A szerző nem járult hozzá nyilatkozatában a dolgozat nyilvánosságra hozásához