THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Online Ramsey Games in Random Graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring proble

A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $\Pi$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $\Omega(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^\epsilon)$, for an arbitrarily small $\epsilon > 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(\beta(k))$, for a function $\beta(\cdot)$, space $O(n^{1+1/k} \cdot \beta)$, and query time $O(n^\epsilon)$, for an arbitrarily small constant $\epsilon >0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $\beta(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+\epsilon,\beta)$-spanners due to Elkin and Peleg. However, so far $(1+\epsilon,\beta)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}

Minimal Ramsey graphs, orthogonal Latin squares, and hyperplane coverings

This thesis consists of three independent parts. The first part of the thesis is concerned with Ramsey theory. Given an integer $q\geq 2$, a graph $G$ is said to be \emph{$q$-Ramsey} for another graph $H$ if in any $q$-edge-coloring of $G$ there exists a monochromatic copy of $H$. The central line of research in this area investigates the smallest number of vertices in a $q$-Ramsey graph for a given $H$. In this thesis, we explore two different directions. First, we will be interested in the smallest possible minimum degree of a minimal (with respect to subgraph inclusion) $q$-Ramsey graph for a given $H$. This line of research was initiated by Burr, Erdős, and Lovász in the 1970s. We study the minimum degree of a minimal Ramsey graph for a random graph and investigate how many vertices of small degree a minimal Ramsey graph for a given $H$ can contain. We also consider the minimum degree problem in a more general asymmetric setting. Second, it is interesting to ask how small modifications to the graph $H$ affect the corresponding collection of $q$-Ramsey graphs. Building upon the work of Fox, Grinshpun, Liebenau, Person, and Szabó and Rödl and Siggers, we prove that adding even a single pendent edge to the complete graph $K_t$ changes the collection of 2-Ramsey graphs significantly. The second part of the thesis deals with orthogonal Latin squares. A {\em Latin square of order $n$} is an $n\times n$ array with entries in $[n]$ such that each integer appears exactly once in every row and every column. Two Latin squares $L$ and $L'$ are said to be {\em orthogonal} if, for all $x,y\in [n]$, there is a unique pair $(i,j)\in [n]^2$ such that $L(i,j) = x$ and $L'(i,j) = y$; a system of {\em $k$ mutually orthogonal Latin squares}, or a {\em $k$-MOLS}, is a set of $k$ pairwise orthogonal Latin squares. Motivated by a well-known result determining the number of different Latin squares of order $n$ log-asymptotically, we study the number of $k$-MOLS of order $n$. Earlier results on this problem were obtained by Donovan and Grannell and Keevash and Luria. We establish new upper bounds for a wide range of values of $k = k(n)$. We also prove a new, log-asymptotically tight, bound on the maximum number of other squares a single Latin square can be orthogonal to. The third part of the thesis is concerned with grid coverings with multiplicities. In particular, we study the minimum number of hyperplanes necessary to cover all points but one of a given finite grid at least $k$ times, while covering the remaining point fewer times. We study this problem for the grid $\mathbb{F}_2^n$, determining the number exactly when one of the parameters $n$ and $k$ is much larger than the other and asymptotically in all other cases. This generalizes a classic result of Jamison for $k=1$. Additionally, motivated by the recent work of Clifton and Huang and Sauermann and Wigderson for the hypercube $\set{0,1}^n\subseteq\mathbb{R}^n$, we study hyperplane coverings for different grids over $\mathbb{R}$, under the stricter condition that the remaining point is omitted completely. We focus on two-dimensional real grids, showing a variety of results and demonstrating that already this setting offers a range of possible behaviors.Diese Dissertation besteht aus drei unabh\"angigen Teilen. Der erste Teil beschäftigt sich mit Ramseytheorie. Für eine ganze Zahl $q\geq 2$ nennt man einen Graphen \emph{$q$-Ramsey} f\"ur einen anderen Graphen $H$, wenn jede Kantenf\"arbung mit $q$ Farben einen einfarbigen Teilgraphen enthält, der isomorph zu $H$ ist. Das zentrale Problem in diesem Gebiet ist die minimale Anzahl von Knoten in einem solchen Graphen zu bestimmen. In dieser Dissertation betrachten wir zwei verschiedene Varianten. Als erstes, beschäftigen wir uns mit dem kleinstm\"oglichen Minimalgrad eines minimalen (bezüglich Teilgraphen) $q$-Ramsey-Graphen f\"ur einen gegebenen Graphen $H$. Diese Frage wurde zuerst von Burr, Erd\H{o}s und Lov\'asz in den 1970er-Jahren studiert. Wir betrachten dieses Problem f\"ur einen Zufallsgraphen und untersuchen, wie viele Knoten kleinen Grades ein Ramsey-Graph f\"ur gegebenes $H$ enthalten kann. Wir untersuchen auch eine asymmetrische Verallgemeinerung des Minimalgradproblems. Als zweites betrachten wir die Frage, wie sich die Menge aller $q$-Ramsey-Graphen f\"ur $H$ verändert, wenn wir den Graphen $H$ modifizieren. Aufbauend auf den Arbeiten von Fox, Grinshpun, Liebenau, Person und Szabó und Rödl und Siggers beweisen wir, dass bereits der Graph, der aus $K_t$ mit einer h\"angenden Kante besteht, eine sehr unterschiedliche Menge von 2-Ramsey-Graphen besitzt im Vergleich zu $K_t$. Im zweiten Teil geht es um orthogonale lateinische Quadrate. Ein \emph{lateinisches Quadrat der Ordnung $n$} ist eine $n\times n$-Matrix, gef\"ullt mit den Zahlen aus $[n]$, in der jede Zahl genau einmal pro Zeile und einmal pro Spalte auftritt. Zwei lateinische Quadrate sind \emph{orthogonal} zueinander, wenn f\"ur alle $x,y\in[n]$ genau ein Paar $(i,j)\in [n]^2$ existiert, sodass es $L(i,j) = x$ und $L'(i,j) = y$ gilt. Ein \emph{k-MOLS der Ordnung $n$} ist eine Menge von $k$ lateinischen Quadraten, die paarweise orthogonal sind. Motiviert von einem bekannten Resultat, welches die Anzahl von lateinischen Quadraten der Ordnung $n$ log-asymptotisch bestimmt, untersuchen wir die Frage, wie viele $k$-MOLS der Ordnung $n$ es gibt. Dies wurde bereits von Donovan und Grannell und Keevash und Luria studiert. Wir verbessern die beste obere Schranke f\"ur einen breiten Bereich von Parametern $k=k(n)$. Zusätzlich bestimmen wir log-asymptotisch zu wie viele anderen lateinischen Quadraten ein lateinisches Quadrat orthogonal sein kann. Im dritten Teil studieren wir, wie viele Hyperebenen notwendig sind, um die Punkte eines endlichen Gitters zu überdecken, sodass ein bestimmter Punkt maximal $(k-1)$-mal bedeckt ist und alle andere mindestens $k$-mal. Wir untersuchen diese Anzahl f\"ur das Gitter $\mathbb{F}_2^n$ asymptotisch und sogar genau, wenn eins von $n$ und $k$ viel größer als das andere ist. Dies verallgemeinert ein Ergebnis von Jamison für den Fall $k=1$. Au{\ss}erdem betrachten wir dieses Problem f\"ur Gitter im reellen Vektorraum, wenn der spezielle Punkt überhaupt nicht bedeckt ist. Dies ist durch die Arbeiten von Clifton und Huang und Sauermann und Wigderson motiviert, die den Hyperwürfel $\set{0,1}^n\subseteq \mathbb{R}^n$ untersucht haben. Wir konzentrieren uns auf zwei-dimensionale Gitter und zeigen, dass schon diese sich sehr unterschiedlich verhalten können

On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. on $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$

Combinatorics and Probability

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers