Problems in Extremal Combinatorics

This dissertation is divided into two major sections. Chapters 1 to 4 are concerned with Turán type problems for disconnected graphs and hypergraphs. In Chapter 5, we discuss an unrelated problem dealing with the equivalence of two notions of stationary processes. The Turán number of a graph H, ex(n,H), is the maximum number of edges in any n-vertex graph which is H-free. We discuss the history and results in this area, focusing particularly on the degenerate case for bipartite graphs. Let Pl denote a path on l vertices, and k*Pl denote k vertex-disjoint copies of Pl. We determine ex(n,k*P3) for n appropriately large, confirming a conjecture of Gorgol. Further, we determine ex(n,k*Pl) for arbitrary l, and n appropriately large. We provide background on the famous Erdös-Sós conjecture, and conditional on its truth we determine ex(n,H) when H is an equibipartite forest, for appropriately large n. In Chapter 4, we prove similar results in hypergraphs. We first discuss the related results for extremal numbers of hyperpaths, before proving the extremal numbers for multiple copies of a loose path of fixed length, and the corresponding result for linear paths. We extend this result to forests of loose hyperpaths, and linear hyperpaths. We note here that our results for loose paths, while tight, do not give the extremal numbers in their classical form; much more detail on this is given in Chapter 4. InChapter 5, we discuss two notions of stationary processes. Roughly, a process is a uniform martingale if it can be approximated arbitrarily well by a process in which the letter distribution depends only on a finite amount of the past. A random Markov process is a process with a coupled `look back\u27 time; that is, to determine the letter distribution, it suffices to choose a random look-back time, and then the distribution depends only on the past up to this time. Kalikow proved that on a binary alphabet, any uniform martingale is also a random Markov process. We extend this result to any finite alphabet

Thresholds for zero-sums with small cross numbers in abelian groups

For an additive group $\Gamma$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $\Gamma$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_\Gamma$. The cross number of $S$ is defined to be the sum $\sum_{i=1}^k 1/|g_i|$, where $|g_i|$ denotes the order of $g_i$ in $\Gamma$. Call $S$ good if it contains a zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved that if $\Gamma$ is abelian then every length $|\Gamma|$ sequence of its elements is good, generalizing a 1989 result of Lemke and Kleitman that had proved an earlier conjecture of Erd\H{o}s and Lemke. In 1989 Chung re-proved the Lemke and Kleitman result by applying a theorem of graph pebbling, and in 2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize Geroldinger's result. Here we use probabilistic theorems from graph pebbling to derive a sharp threshold version of Geroldinger's theorem for abelian groups of a certain form. Specifically, we prove that if $p_1, \ldots, p_d$ are (not necessarily distinct) primes and $\Gamma_k$ has the form $\prod_{i=1}^d {\mathbb Z}_{p_i^k}$ then there is a function $\tau=\tau(k)$ (which we specify in Theorem 4) with the following property: if $t-\tau\rightarrow\infty$ as $k\rightarrow\infty$ then the probability that $S$ is good in $\Gamma_k$ tends to 1, while if $\tau-t\rightarrow\infty$ then that probability tends to 0

Thresholds for Pebbling on Grids

Given a connected graph $G$ and a configuration of $t$ pebbles on the vertices of G, a $q$-pebbling step consists of removing $q$ pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector $\bf{q}=(q_1,\ldots,q_d)$, $\bf{q}$-pebbling consists of allowing $q_i$-pebbling in coordinate $i$. A distribution of pebbles is called solvable if it is possible to transfer at least one pebble to any specified vertex of $G$ via a finite sequence of pebbling steps. In this paper, we determine the weak threshold for $\bf{q}$-pebbling on the sequence of grids $[n]^d$ for fixed $d$ and $\bf{q}$, as $n\to\infty$. Further, we determine the strong threshold for $q$-pebbling on the sequence of paths of increasing length. A fundamental tool in these proofs is a new notion of centrality, and a sufficient condition for solvability based on the well used pebbling weight functions; we believe this weight lemma to be the first result of its kind, and may be of independent interest. These theorems improve recent results of Czygrinow and Hurlbert, and Godbole, Jablonski, Salzman, and Wierman. They are the generalizations to the random setting of much earlier results of Chung. In addition, we give a short counterexample showing that the threshold version of a well known conjecture of Graham does not hold. This uses a result for hypercubes due to Czygrinow and Wagner.Comment: 16 pages; comments are welcom

Tur\`an numbers of Multiple Paths and Equibipartite Trees

The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.Comment: 17 pages, 13 figures; Updated to incorporate referee's suggestions; minor structural change

Random-step Markov processes

We explore two notions of stationary processes. The first is called a random-step Markov process in which the stationary process of states, $(X_i)_{i \in \mathbb{Z}}$ has a stationary coupling with an independent process on the positive integers, $(L_i)_{i \in \mathbb{Z}}$ of `random look-back distances'. That is, $L_0$ is independent of the `past states', $(X_i, L_i)_{i<0}$, and for every positive integer $n$, the probability distribution on the `present', $X_0$, conditioned on the event $\{L_0 = n\}$ and on the past is the same as the probability distribution on $X_0$ conditioned on the `$n$-past', $(X_i)_{-n\leq i <0}$ and $\{L_0 = n\}$. A random Markov process is a generalization of a Markov chain of order $n$ and has the property that the distribution on the present given the past can be uniformly approximated given the $n$-past, for $n$ sufficiently large. Processes with the latter property are called uniform martingales, closely related to the notion of a `continuous $g$-function'. We show that every stationary process on a countable alphabet that is a uniform martingale and is dominated by a finite measure is also a random Markov process and that the random variables $(L_i)_{i \in \mathbb{Z}}$ and associated coupling can be chosen so that the distribution on the present given the $n$-past and the event $\{L_0 = n\}$ is `deterministic': all probabilities are in $\{0,1\}$. In the case of finite alphabets, those random-step Markov processes for which $L_0$ can be chosen with finite expected value are characterized. For stationary processes on an uncountable alphabet, a stronger condition is also considered which is sufficient to imply that a process is a random Markov processes. In addition, a number of examples are given throughout to show the sharpness of the results.Comment: 31 page