Combinatorics and Metric Geometry

Abstract

This thesis consists of an introduction and seven chapters, each devoted to a different combinatorial problem.

In Chapters 1 and 2, we consider the main subject of this thesis; the sharp stability of the Brunn-Minkowski inequality (BM). This celebrated theorem from the 19th century asserts that for bodies A,B $\subset$ $\mathbb{R}$$^{k}$, we have

|A + B|$^{1/k}$ $\geq$ |A|$^{1/k}$ + |B|$^{1/k}$,

where |$\cdot$| is the Lebesgue measure and A + B := {a + b : a $\in$ A, b $\in$ B} is the Minkowski sum. Moreover, we have equality if and only if A,B are homothetic convex sets. The stability question, studied in many papers, asks how the distance to equality in BM relates to the distance from A,B to homothetic convex sets. In particular, given Brunn-Minkowsi deficit

$\delta$ := |A+B|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$ -1,

and normalized volume ratio

$\textit{t}$ := |A|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$,

what is the best bound one can find on

$\omega$ := |K$_{A}$ \ A| / |A| + |K$_{B}$ \ B| / |B|,

where K$_{A}$ $\supset$ A, and K$_{B}$ $\supset$ B are homothetic convex sets of minimal size? In Chapter 2, we prove a conjecture by Figalli and Jerison establishing the sharp stability for homothetic sets. In particular, we show that for homothetic sets, we have $\omega$ = O$_{k}$($\delta$t$^{-1}$), for $\delta$ sufficiently small. In Chapter 3, we establish the sharp stability for planar sets, i.e. we show that for planar sets and $\delta$ sufficiently small, we have $\omega$ = O($\delta$$^{1/2}$t$^{-1/2}$). A crucial result in Chapter 3 shows that for any $\epsilon$ > 0, if $\delta$ is sufficiently small, then we have

|co(A + B) \ (A + B)| $\leq$ (1 + $\epsilon$)(|co(A) \ A| + |co(B) \ B|).

In Chapter 4, we consider a reconstruction problem for functions on graphs. Given a function $\textit{f}$:V(G) $\rightarrow$ [k] on the vertices of a graph G and a random walk (U$_{i}$)$^{{\infty}}_{i = 1}$ on that graph, can we reconstruct $\textit{f}$ (up to automorphisms) based on just ($\textit{f}$(U$_{i}$)$^{{\infty}}_{i = 1}$? Gross and Grupel showed this was not generally possible on the hypercube, by constructing non-isomorphic $\textit{locally p-biased}$ sets $\textit{X}$, so that for each vertex $\textit{v}$ the fraction of neighbours which is in $\textit{X}$ is exactly $\textit{p}$. Answering a question of Gross and Grupel, we construct uncountably many non-isomorphic partitions of $\mathbb{Z}$$^{k}$ into 2k parts such that every element of $\mathbb{Z}$$^{k}$ has exactly one neighbour in each part. As a result, we find $\textit{locally p-biased}$ sets for all $\textit{p = c/2n}$ with $\textit{c}$ $\in$ {0, ... , 2n}.

In Chapter 5, we prove the complete graph case of the bunkbed conjecture. Given a graph G, let the bunkbed graph BB(G) be the graph G$\Box$K$_{2}$, i.e. the graph obtained from considering two copies of G and connecting equivalent vertices with an edge. The bunkbed conjecture posed by Kasteleyn in 1985 asserts the very intuitive statement that when considering percolation with uniform parameter p, we have $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{1}$) $\geq$ $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{2}$), i.e. a vertex has a higher probability of being connected to a vertex in the same copy of G than being connected to the equivalent vertex in the other copy of G.

In Chapter 6, we consider the (t,r) broadcast domination number, a generalisation of the domination number in graphs. In this form of domination, we consider a set T $\subset$ V(G) of towers which broadcast at strength t, where broadcast strength decays linearly with distance in the graph. A set of towers is (t,r) broadcast dominating if every vertex in the graph receives at least r signal from all towers combined. More formally, the (t,r) broadcast domination number of a graph G is the minimal cardinality of a set T $\subset$ V(G) such that for every vertex v $\in$ V(G), we have

$^{{\Sigma}}_{u {{\in}} T}$ max{t - d(u,v),0} $\geq$ r.

Proving a conjecture by Drews, Harris, and Randolph, we establish that the minimal asymptotical density of (t,3) broadcasting subset of $\mathbb{Z}$$^{2}$ is the same as the minimal asymptotical density of a (t-1,1) broadcasting subset of $\mathbb{Z}$$^{2}$.

In Chapter 7, we consider the eternal game chromatic number, a version of the game chromatic number in which the game continues after all vertices have been coloured. We show that with high probability

$\chi$$^\infty_g$ (G$_{n,p}$) = (p/2 + o(1))n for odd n, and also for even n when p = 1/k for some k $\in$ $\mathbb{N}$. The upper bound applies for even n and any other value of p as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza.

In Chapter 8, we consider the bridge-burning cops and robbers game, a version of the game where after a robber moves over an edge, the edge is removed from the graph. Proving a generalization of a conjecture by Kinnersley and Peterson, we establish the asymptotically maximal capture time in this game for graphs with bridge-burning cops number at least three. In particular, we show that this maximal capture time grows as

k$^{-O(k)}$n$^{k+2}$, where k $\geq$ 3 is the bridge burning cop number and n is the number of vertices of the graph.