Half Covering, Half Coloring Open Access

Clifton, Alexander (Spring 2022)

Permanent URL: https://etd.library.emory.edu/concern/etds/k35695826?locale=en


Alon and F\"{u}redi determined the minimum number of affine hyperplanes needed to cover all but one point of an $n$-dimensional rectangular grid. It is natural to extend this to higher covering multiplicities and ask for the minimum number of affine hyperplanes needed to cover every grid point at least $k$ times each, except for one point that is not covered at all. In the special case of an $n$-cube, we use a Punctured Combinatorial Nullstellensatz of Ball and Serra to exactly determine the minimum for $k=3$ and to formulate a lower bound for $k\geq{4}$. We also treat the problem as an integer program to determine an asymptotic answer for fixed $n$ as $k\to\infty$. Again using the Punctured Combinatorial Nullstellensatz, we answer the question for a general grid when $k=2$. 

Another generalization we address for the $n$-cube is the minimum number of affine subspaces of codimension $d$ needed to cover all but one vertex at least once. We also consider the minimum number of hyperplanes needed to cover all points of a triangular grid.

In the second half of this dissertation, we consider arithmetic Ramsey theory problems in the spirit of van der Waerden's theorem. For a set $D\subset\mathbb{Z}_{>0}$, Landman and Robertson introduced the notion of a $D$-diffsequence, which is an increasing sequence $a_1<\cdots

By considering the case where $D$ consists of all powers of $2$, we provide an example of a $2$-accessible set where $\Delta(D,k;2)$ grows faster than any polynomial. The proof relies on a series of periodic colorings based on the Thue--Morse sequence. We also use Beatty sequences to classify which sets of the form $D=\{d_1,d_2,\cdots\}$ with $d_i\mid d_{i+1}$ for all $i$ are $2$-accessible.

Table of Contents

1 Introduction

1.1 Covering and Coloring

1.2 Background for Covering

1.2.1 Covering Problems

1.2.2 Combinatorial Nullstellensatz

1.3 Background for Coloring

1.3.1 Ramsey Theory

1.3.2 Diffsequences

1.4 Dissertation Synopsis

2 Covering for Hypercubes

2.1 Preliminary Results

2.2 Fixed Covering Multiplicity

2.3 Fixed Dimension

2.4 Higher Codimension

2.5 Connections to Graph Covering and Sidon Sets

3 Covering for General Grids

3.1 Rectangular Grids

3.1.1 Complications for k=3

3.1.2 Restrictions on the Minimum Degree Polynomial

3.2 Triangular Grids

3.2.1 With a Point Uncovered

3.2.2 With All Points Covered

4 Results in Arithmetic Ramsey Theory

4.1 Superpolynomial Growth for Delta(D,k;2)

4.2 Inaccessibility of Certain Sets

4.2.1 Factorials

4.2.2 Existence Proof for General Dividing Sets

4.2.3 Random Diffsequences

A Number of Lines Needed to Cover Triangular Grids

B Values of Delta(D,k;2) when D={2^i | i is a nonnegative integer}

C Gurobi Code


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