arXiv subjects

Conner Griffin

Publications and source records attributed to Conner Griffin.

3 recordsLinked to original sources

arXiv2026

Infinite Sum-Product Configurations in Parallel

We show that for any finite partition of $\mathbb{N}$ there is an infinite sequence whose finite sums are monochromatic and such that infinitely many of the products with a fixed number of factors are monochromatic -- though not necessarily belonging to the same color class as the finite sums. We are able to build these infinite configurations in parallel by refining arbitrary partitions of $\mathbb{N}$. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of $\mathbb{N}$.

math.CO↗

arXiv2025

Characterizing $ (\mathcal{F}, \mathcal{G}) $-syndetic, $ (\mathcal{F}, \mathcal{G}) $-thick, and related notions of size using derived sets along ultrafilters

We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-\v{C}ech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-\v{C}ch compactification, our particular use of derived sets adapt and generalize methods recently used by Griffin arXiv:2311.09436 to characterize relative piecewise syndetic sets. As an application, we define an algebraically interesting subset of the Stone-\v{C}ech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal.

math.GN↗

arXiv2023

A characterization of piecewise $\mathcal{F}$-syndetic sets

Some filter relative notions of size, $\left( \mathcal{F},\mathcal{G}\right) $-syndeticity and piecewise $\mathcal{F} $-syndeticity, were defined and applied with clarity and focus by Shuungula, Zelenyuk and Zelenyuk in their paper ``The closure of the smallest ideal of an ultrafilter semigroup.'' These notions are generalizations of the well studied notions of syndeticity and piecewise syndeticity. Since then, there has been an effort to develop the theory around the algebraic structure of the Stone-\v{C}ech compactification so that it encompasses these new generalizations. In this direction, we prove a characterization of piecewise $\mathcal{F}$-syndetic sets. This fully answers a conjecture of Christopherson and Johnson. arXiv:2105.09723

math.GN↗