Date of Award
1-1-2025
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
First Advisor
Thái Hoàng Lê
Second Advisor
Thái Hoàng Lê
Third Advisor
Ayla Gafni
School
University of Mississippi
Relational Format
dissertation/thesis
Abstract
Arithmetic Ramsey theory is an area of combinatorics that deals with finding structures in large subsets of integers, dating back to at least Hilbert (1892). One of the early results in this area is Schur’s theorem (1917) that pioneered the concept of partition regularity. In 1933, Schur’s student Rado generalized his theorem and proposed a conjecture for partition regular equations. This was settled by Deuber in 1973. The first part of this dissertation is to generalize this conjecture in infinite integral domains.
For non-partition regular equations Rado gave a Boundedness conjecture in the integers. While this conjecture is still open, Bergelson, Deuber, Hindman and Lefmann (1994) provided a counterexample to show that this conjecture does not hold true in arbitrary commutative rings. In the second part of this dissertation, we provide a generalization of this counterexample.
Another branch of Arithmetic Ramsey theory deals with density results. Sárközy theorem is a central result in this area. In 2013, Lê and Liu proved a function field analogue of Sárközy’s theorem. Further, in 2016, Green improved their result using the Croot-Lev- Pach polynomial method. In the third part of this dissertation, we generalize Green’s theorem using the Croot-Lev-Pach polynomial method and the concept of slice rank introduced by Tao. This is joint work with Thái Hoàng Lê and Pierre–Yves Bienvenu.
Recommended Citation
Wathodkar, Gauree, "Arithmetic Ramsey Theory in Algebraic Settings" (2025). Electronic Theses and Dissertations. 3405.
https://egrove.olemiss.edu/etd/3405