Conner Griffin
The Ohio State University
Title
Filter quantifiers, algebra of collections, and applications to Ramsey theory
Abstract
Ultrafilters on the positive integers can be defined as those quantifiers which preserve conjunction, disjunction, and negation of formulas in a single free variable. Filters are those quantifiers which preserve conjunction. In the case of non-maximal filters, we need a distinct quantifier to establish relations with disjunction and negation. As an example, the Frechet filter on the natural numbers - which is typically taken to be the collection of all cofinite subsets of the naturals - is the “all but a finite number of” quantifier. This quantifier does not preserve negation or disjunction, and so we complement it with the “for infinitely many” quantifier. This is the same relationship shared by the universal and existential quantifiers. For these quantifiers on a semigroup, I will define an associative binary operation. This operation allows us to identify sets which are large in a wide range of possible algebraic ways. I will present simple examples demonstrating how these notions of size naturally lead to interesting variations of known results from semigroup Ramsey theory, such as Hindman’s finite sums theorem and van der Warden’s theorem on arithmetic progressions.