Electronic Theses and Dissertations

Date

2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

Randall McCutcheon

Committee Member

Alistair Windsor

Committee Member

David Grynkiewicz

Committee Member

Mate Wierdl

Abstract

There are many branches of problems in Ramsey theory which each belong to at least one of a number of subjects: graph theory, number theory and set theory. The successful methods of solution are also varied. Methods of solution have been found relying on probability theory, ergodic theory, dynamical systems theory, algebra and topology. We take the pigeonhole principle as a foundation for the problems in Ramsey theory that we consider. Pigeonhole Principle. Let A be a set with |A| > n for some n ∈ N. Let A = Sn i=1 Ci be a finite partition of A. There is an i ∈ [n] such that |Ci| ≥ 2. We can generalize the pigeonhole principle in number theoretic directions. In such cases, one would like to find a large, complex and sparse algebraic configuration in some cell of any finite partition of N or a more general semigroup. There are two historically noteworthy examples. The first is van der Waerden’s theorem which is one of the earliest results in Ramsey theory. The algebraic configuration which is sought are arbitrarily long arithmetic progressions – affine images of discrete intervals. Van der Waerden’s Theorem. [32] Let N = Sn i=1 Ci be a finite partition. Then there exists an i ∈ [n] such that for all k ∈ N there exist a, b ∈ N such that {a, a + b, a + 2b, . . . , a + (k − 1) b} ⊂ Ci. A second noteworthy example is Hindman’s theorem. Here the algebraic configuration are finite sums over a sequence. There is a stronger version for arbitrary semigroups which can be seen in [24, Corollary 5.9] or in Chapter 1 here as Theorem 1.60. Hindman’s Theorem. [20] Let N = Sn i=1 Ci be a finite partition. Then there exists an iv i ∈ [n] and a sequence (xj )∞ j=1 such that FS ? (xj )∞ j=1 ? ⊆ Ci. The approaches to such problems which appear here rely on ergodic theory, topological dynamics, general combinatorial methods, and especially the algebraic structure of the Stone- ˇCech compactification of a discrete semigroup. The first chapter introduces the Stone- ˇCech compactification of a discrete semigroup as a topological semigroup of ultrafilters. That the Stone- ˇCech compactification of a discrete space, X, is homeomorphic to a topological space of all ultrafilters on X is established in greater generality through the Wallman compactification. The necessary details about the semigroup structure of the Stone- ˇCech compactification is then discussed. Also in this chapter is the basic ergodic theoretric material needed in chapter four. We close out the introduction with a survey of combinatorial results which provide context for the results in chapters two through four. The second chapter is an in-depth analysis of filter relative notions of size in a semigroup. Research interest in this direction was initiated by Shuungula, Zelenyuk, and Zelenyuk in [31]. Combinatorial applications of these broad notions are demonstrated in [17, 25, 2, 4, 22, 29]. The last four predate the paper of Shuungula, Zelenyuk, and Zelenyuk. As a result of this analysis, a new perspective of several notions of size is introduced. This new perspective is then used to establish a characterization of filter relative piecewise syndetic sets. In the third chapter, filter relative notions of size are successfully applied to some com- binatorial problems. This includes a combinatorial characterization of filter relative central sets. In addition filter relative central sets are used to prove an infinitary nilpotent polyno- mial Hales-Jewett theorem from the finitary nilpotent polynomial Hales-Jewett theorem of Johnson and Richter [25]. The fourth chapter contains some partial progress towards a multidimensional Szemer´edi v theorem for the finitely generated free semigroup. This work is based on the methods of Bulinski and Fish in [7]

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