- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Extremal problems in combinatorics and geometry
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Extremal problems in combinatorics and geometry Currier, Gabriel
Abstract
This thesis is comprised of four chapters relating to combinatorics and geometry. More specifically, the main topics of the dissertation are incidence geometry and Euclidean Ramsey theory. In Chapter 2, we study the Erdős unit distance problem. In particular, we prove a structural result for pointsets determining many unit distances from a small direction set in any norm. Then, we use this to deduce that, in the Euclidean norm, such pointsets cannot exist. In chapter 3, we give new sharp constructions for the Szemerédi-Trotter problem, where the coordinates of the points come from arbitrary number fields. This adds to previous constructions of Erdős, Elekes, Sheffer and Silier, and Guth and Silier. In Chapter 4, we study a problem in Euclidean Ramsey theory. In particular, we show that in any two-coloring of the plane and positive real number α, there exists a monochromatic three-term arithmetic progression with common distance α. This also proves a significant unsolved case in a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus. In Chapter 5, we show some negative results in Euclidean Ramsey Theory. In particular, we give red/blue colorings of n-dimensional Euclidean space that avoid short arithmetic progressions in red and long ones in blue. This continues a recent line of work of Conlon and Wu and Führer and Tóth.
Item Metadata
Title |
Extremal problems in combinatorics and geometry
|
Creator | |
Supervisor | |
Publisher |
University of British Columbia
|
Date Issued |
2025
|
Description |
This thesis is comprised of four chapters relating to combinatorics and geometry. More specifically, the main topics of the dissertation are incidence geometry and Euclidean Ramsey theory.
In Chapter 2, we study the Erdős unit distance problem. In particular, we prove a structural result for pointsets determining many unit distances from a small direction set in any norm. Then, we use this to deduce that, in the Euclidean norm, such pointsets cannot exist.
In chapter 3, we give new sharp constructions for the Szemerédi-Trotter problem, where the coordinates of the points come from arbitrary number fields. This adds to previous constructions of Erdős, Elekes, Sheffer and Silier, and Guth and Silier.
In Chapter 4, we study a problem in Euclidean Ramsey theory. In particular, we show that in any two-coloring of the plane and positive real number α, there exists a monochromatic three-term arithmetic progression with common distance α. This also proves a significant unsolved case in a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus.
In Chapter 5, we show some negative results in Euclidean Ramsey Theory. In particular, we give red/blue colorings of n-dimensional Euclidean space that avoid short arithmetic progressions in red and long ones in blue. This continues a recent line of work of Conlon and Wu and Führer and Tóth.
|
Genre | |
Type | |
Language |
eng
|
Date Available |
2025-07-21
|
Provider |
Vancouver : University of British Columbia Library
|
Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
|
DOI |
10.14288/1.0449458
|
URI | |
Degree (Theses) | |
Program (Theses) | |
Affiliation | |
Degree Grantor |
University of British Columbia
|
Graduation Date |
2025-11
|
Campus | |
Scholarly Level |
Graduate
|
Rights URI | |
Aggregated Source Repository |
DSpace
|
Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International