arXiv subjects

John H. Johnson Jr.

Publications and source records attributed to John H. Johnson Jr..

2 recordsLinked to original sources

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-Čech compactification and its applications. Combined with the existence of idempotents and structure of the smallest ideal in closed subsemigroups of the Stone-Čch 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-Čech compactification and show, in some ways, it shares structural properties analogous to the smallest ideal.

math.GN↗

arXiv2016

New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.

math.CO↗