Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, every $k$-latin square of order $n$ is separable. We discuss the connection between $k$-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and $k$-latin trades. We also enumerate and classify $k$-latin squares of small orders.Comment: Final version as sent to journa

The Quasigroup Block Cipher and its Analysis

This thesis discusses the Quasigroup Block Cipher (QGBC) and its analysis. We first present the basic form of the QGBC and then follow with improvements in memory consumption and security. As a means of analyzing the system, we utilize tools such as the NIST Statistical Test Suite, auto and crosscorrelation, then linear and algebraic cryptanalysis. Finally, as we review the results of these analyses, we propose improvements and suggest an algorithm suitable for low-cost FPGA implementation

Channel Access Management in Data Intensive Sensor Networks

There are considerable challenges for channel access in Data Intensive Sensor Networks - DISN, supporting Data Intensive Applications like Structural Health Monitoring. As the data load increases, considerable degradation of the key performance parameters of such sensor networks is observed. Successful packet delivery ratio drops due to frequent collisions and retransmissions. The data glut results in increased latency and energy consumption overall. With the considerable limitations on sensor node resources like battery power, this implies that excessive transmissions in response to sensor queries can lead to premature network death. After a certain load threshold the performance characteristics of traditional WSNs become unacceptable. Research work indicates that successful packet delivery ratio in 802.15.4 networks can drop from 95% to 55% as the offered network load increases from 1 packet/sec to 10 packets/sec. This result in conjunction with the fact that it is common for sensors in an SHM system to generate 6-8 packets/sec of vibration data makes it important to design appropriate channel access schemes for such data intensive applications.In this work, we address the problem of significant performance degradation in a special-purpose DISN. Our specific focus is on the medium access control layer since it gives a fine-grained control on managing channel access and reducing energy waste. The goal of this dissertation is to design and evaluate a suite of channel access schemes that ensure graceful performance degradation in special-purpose DISNs as the network traffic load increases.First, we present a case study that investigates two distinct MAC proposals based on random access and scheduling access. The results of the case study provide the motivation to develop hybrid access schemes. Next, we introduce novel hybrid channel access protocols for DISNs ranging from a simple randomized transmission scheme that is robust under channel and topology dynamics to one that utilizes limited topological information about neighboring sensors to minimize collisions and energy waste. The protocols combine randomized transmission with heuristic scheduling to alleviate network performance degradation due to excessive collisions and retransmissions. We then propose a grid-based access scheduling protocol for a mobile DISN that is scalable and decentralized. The grid-based protocol efficiently handles sensor mobility with acceptable data loss and limited overhead. Finally, we extend the randomized transmission protocol from the hybrid approaches to develop an adaptable probability-based data transmission method. This work combines probabilistic transmission with heuristics, i.e., Latin Squares and a grid network, to tune transmission probabilities of sensors, thus meeting specific performance objectives in DISNs. We perform analytical evaluations and run simulation-based examinations to test all of the proposed protocols

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

An approximate version of a conjecture of Aharoni and Berger

Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by n colours with at least n+1 edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the AharoniBerger Conjecture is proved—it is shown that if there are at least n + o(n) edges of each colour in a proper n-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour

Spatio-Causal Patterns of Sample Growth

Different statistical samples (e.g., from different locations) offer populations and learning systems observations with distinct statistical properties. Samples under (1) 'Unconfounded' growth preserve systems' ability to determine the independent effects of their individual variables on any outcome-of-interest (and lead, therefore, to fair and interpretable black-box predictions). Samples under (2) 'Externally-Valid' growth preserve their ability to make predictions that generalize across out-of-sample variation. The first promotes predictions that generalize over populations, the second over their shared uncontrolled factors. We illustrate these theoretic patterns in the full American census from 1840 to 1940, and samples ranging from the street-level all the way to the national. This reveals sample requirements for generalizability over space and time, and new connections among the Shapley value, counterfactual statistics, and hyperbolic geometry

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

Problems and results on linear hypergraphs

In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint. We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an $n\times n$ Latin square $A$ one can define a binary operation $\circ:[n]\times[n]\to [n]$, and $\circ$ is associative if and only if $A$ is a group multiplication table. Hrushovski asked whether, if $\circ$ is only associative a positive proportion of the time, $A$ must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99$\%$' version and then to a `1$\%$' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within $A$ a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory. We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square $A$, proves that $A$ is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of $\circ$ are far more complex than those for associativity, topological complications arise that are not present in the earlier work. We also study a problem of Loh concerning sequences of triples of integers from $[n]$ satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of $n^2$ to $n^2/\exp(\log^*n)$ using the triangle removal lemma and conjecturing that a natural construction of length $n^{3/2}$ is best possible. We provide the first power-type improvement to the upper bound, showing that there exists $\epsilon>0$ such that the length is bounded by $n^{2-\epsilon}$. By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture. Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in $\mathbb{Z}_{2^n}$, and resolve a conjecture concerning a diffusion process on graphs