Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics

Abstract

In this thesis, we consider several combinatorial topics, belonging to the areas appearing in the thesis title. Given a non-empty complete metric space $(X,d)$, a family of $n$ continuous maps $f_1,f_2,\dots,f_n\colon X\to X$ is a \emph{contractive family} if there exists $\lambda0$ and $k\in\mathbb{N}$, we construct a subset $A\subset\mathbb{Z}_N$ for some $N$, such that $|A^2+kA|\leq\epsilon N$, while $A-A=\mathbb{Z}_N$. (Here $A-A=\{a_1-a_2:a_1,a_2\in A\}$ and $A^2+kA=\{a_1a_2+a'_1+a'_2+\dots+a'_k:a_1,a_2,a'_1,a'_2,\dots,a'_k\in A\}$.) We also prove some extensions of this result. Among other ingredients, the proof also includes an application of a quantitative equidistribution result for polynomials. In the final part, we consider the Graham-Pollak problem for hypergraphs. Let $f_r(n)$ be the minimum number of complete $r$-partite $r$-graphs needed to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. We disprove a conjecture that $f_4(n)\geq (1+o(1))\binom{n}{2}$, by showing that $f_4(n)\leq\frac{14}{15}(1+o(1))\binom{n}{2}$. The proof is based on the relationship between this problem and a problem about decomposing products of complete graphs, and understanding how the Graham-Pollak theorem (for graphs) affects what can happen here.I would like to thank Trinity College and Department of Pure Mathematics and Mathematical Statistics for their generous financial support and hospitality during PhD studies