Composing some notions of largeness and a classification problem for certain structured sets


Syndetic and thick subsets of a discrete semigroup $S$ are two well-known (and classical) "notions of largeness" that have nice algebraic characterizations in $\beta S$ related to the smallest ideal $K(\beta S)$. Moreover, the connections between these notions of largeness and van der Waerden's theorem on arithmetic progressions are among some of the earliest results in Ramsey Theory.

We'll state generalizations for both syndetic and thick subsets and show how these notions of largeness can be "composed" to produce different (and possibly new) notions of largeness. (The generalization of syndetic subset that we state is due to Shuungula, Zelenyuk, and Zelenyuk, but this notion has appeared, more or less, implicitly if unnamed in the literature connected to the algebraic structure of $\beta S$.) Some of these composite notions are well known, such as piecewise syndetic sets. Other composite notions are trivial, reducing to the collection $\{S\}$. While a few others appear to be new.

We'll sketch a suggestive visualization of these notions, outline the proof of some characterizations of these generalizations, under certain relatively mild conditions, formulate a classification problem, and outline some connections to Ramsey Theory.

Sat, Nov 2, 2019 12:00 AM — 6:38 PM
University of Florida (Little Hall Room 127)
Gainesville, FL