In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalised by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of the Littlewood Offord result, a sharp anti-concentration inequality for products of independent random variables. (C) 2016 Elsevier Inc. All rights reserved.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

In this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}. / Master of Science / In this thesis we consider the following question: Given a number of triangles, t, where each of these triangles are different, we ask what is the maximum number of points that can be placed in d-dimensional space, such that every triplets of these points form the vertices of only the t allowable triangles. We answer this for every dimension, d when the number of triangles is t = 1, as well as show that when t = 2 triangle are in d = 3-dimensional space. This set of questions rises from considering the work of Erd˝os and Fishburn in higher dimensional space [EF].

The problems we study in this thesis arise in computer science, extremal set theory and quantum computing. The first common feature of these problems is that each can be reduced to characterizing the independent sets of maximum size in a suitable graph. A second common feature is that the size of these independent sets meets an eigenvalue bound due to Delsarte and Hoffman. Thirdly, the graphs that arise belong to association schemes that have already been studied in other contexts. Our first problem involves covering arrays on graphs, which arises in computer science. The goal is to find a smallest covering array on a given graph <i>G</i>. It is known that this is equivalent to determining whether <i>G</i> has a homomorphism into a <i>covering array graph</i>, <i>CAG(n,g)</i>. Thus our question: Are covering array graphs cores? A covering array graph has as vertex set the partitions of <i>{1,. . . ,n}</i> into <i>g</i> cells each of size at least <i>g</i>, with two vertices being adjacent if their meet has size <i>g²</i>. We determine that <i>CAG(9,3)</i> is a core. We also determine some partial results on the family of graphs <i>CAG(g²,g)</i>. The key to our method is characterizing the independent sets that meet the Delsarte-Hoffman bound---we call these sets <i>ratio-tight</i>. It turns out that <i>CAG(9,3)</i> sits inside an association scheme, which will be useful but apparently not essential. We then turn our attention to our next problem: the Erdos-Ko-Rado theorem and its <i>q</i>-analogue. We are motivated by a desire to find a unifying proof that will cover both versions. The EKR theorem gives the maximum number of pairwise disjoint <i>k</i>-sets of a fixed <i>v</i>-set, and characterizes the extremal cases. Its <i>q</i>-analogue does the same for <i>k</i>-dimensional subspaces of a fixed <i>v</i>-dimensional space over <i>GF(q)</i>. We find that the methods we developed for covering array graphs apply to the EKR theorem. Moreover, unlike most other proofs of EKR, our argument applies equally well to the <i>q</i>-analogue. We provide a proof of the characterization of the extremal cases for the <i>q</i>-analogue when <i>v=2k</i>; no such proof has appeared before. Again, the graphs we consider sit inside of well-known association schemes; this time the schemes play a more central role. Finally, we deal with the problem in quantum computing. There are tasks that can be performed using quantum entanglement yet apparently are beyond the reach of methods using classical physics only. One particular task can be solved classically if and only if the graph &Omega;(<i>n</i>) has chromatic number <i>n</i>. The graph &Omega;(<i>n</i>) has as vertex set the set of all <i>?? 1</i> vectors of length <i>n</i>, with two vertices adjacent if they are orthogonal. We find that <i>n</i> is a trivial upper bound on the chromatic number, and that this bound holds with equality if and only if the Delsarte-Hoffman bound on independent sets does too. We are thus led to characterize the ratio-tight independent sets. We are then able to leverage our result using a recursive argument to show that <i>&chi;</i>(&Omega;(<i>n</i>)) > <i>n</i> for all <i>n</i> > 8. It is notable that the reduction to independent sets, the characterization of ratio-tight sets, and the recursive argument all follow from different proofs of the Delsarte-Hoffman bound. Furthermore, &Omega;(<i>n</i>) also sits inside a well-known association scheme, which again plays a central role in our approach.

Estudamos o problema de famílias intersectantes extremais em um subconjunto aleatório da família dos subconjuntos com exatamente k elementos de um conjunto dado. Obtivemos uma descrição quase completa da evolução do tamanho de tais famílias. Versões semelhantes do problema foram estudadas por Balogh, Bohman e Mubayi em 2009, e por Hamm e Kahn, e Balogh, Das, Delcourt, Liu e Sharifzadeh de maneira concorrente a este trabalho. / We studied the problem of maximal intersecting families in a random subset of the family of subsets with exactly k elements from a given set. We obtained a nearly complete description of the evolution of the size of such families. Similar versions of this problem have been studied by Balogh, Bohman and Mubayi in 2009, and by Hamm and Kahn, and Balogh, Das, Delcourt, Liu and Sharifzadeh concurrently with this work.

The problems we study in this thesis arise in computer science, extremal set theory and quantum computing. The first common feature of these problems is that each can be reduced to characterizing the independent sets of maximum size in a suitable graph. A second common feature is that the size of these independent sets meets an eigenvalue bound due to Delsarte and Hoffman. Thirdly, the graphs that arise belong to association schemes that have already been studied in other contexts. Our first problem involves covering arrays on graphs, which arises in computer science. The goal is to find a smallest covering array on a given graph <i>G</i>. It is known that this is equivalent to determining whether <i>G</i> has a homomorphism into a <i>covering array graph</i>, <i>CAG(n,g)</i>. Thus our question: Are covering array graphs cores? A covering array graph has as vertex set the partitions of <i>{1,. . . ,n}</i> into <i>g</i> cells each of size at least <i>g</i>, with two vertices being adjacent if their meet has size <i>g²</i>. We determine that <i>CAG(9,3)</i> is a core. We also determine some partial results on the family of graphs <i>CAG(g²,g)</i>. The key to our method is characterizing the independent sets that meet the Delsarte-Hoffman bound---we call these sets <i>ratio-tight</i>. It turns out that <i>CAG(9,3)</i> sits inside an association scheme, which will be useful but apparently not essential. We then turn our attention to our next problem: the Erdos-Ko-Rado theorem and its <i>q</i>-analogue. We are motivated by a desire to find a unifying proof that will cover both versions. The EKR theorem gives the maximum number of pairwise disjoint <i>k</i>-sets of a fixed <i>v</i>-set, and characterizes the extremal cases. Its <i>q</i>-analogue does the same for <i>k</i>-dimensional subspaces of a fixed <i>v</i>-dimensional space over <i>GF(q)</i>. We find that the methods we developed for covering array graphs apply to the EKR theorem. Moreover, unlike most other proofs of EKR, our argument applies equally well to the <i>q</i>-analogue. We provide a proof of the characterization of the extremal cases for the <i>q</i>-analogue when <i>v=2k</i>; no such proof has appeared before. Again, the graphs we consider sit inside of well-known association schemes; this time the schemes play a more central role. Finally, we deal with the problem in quantum computing. There are tasks that can be performed using quantum entanglement yet apparently are beyond the reach of methods using classical physics only. One particular task can be solved classically if and only if the graph &Omega;(<i>n</i>) has chromatic number <i>n</i>. The graph &Omega;(<i>n</i>) has as vertex set the set of all <i>± 1</i> vectors of length <i>n</i>, with two vertices adjacent if they are orthogonal. We find that <i>n</i> is a trivial upper bound on the chromatic number, and that this bound holds with equality if and only if the Delsarte-Hoffman bound on independent sets does too. We are thus led to characterize the ratio-tight independent sets. We are then able to leverage our result using a recursive argument to show that <i>&chi;</i>(&Omega;(<i>n</i>)) > <i>n</i> for all <i>n</i> > 8. It is notable that the reduction to independent sets, the characterization of ratio-tight sets, and the recursive argument all follow from different proofs of the Delsarte-Hoffman bound. Furthermore, &Omega;(<i>n</i>) also sits inside a well-known association scheme, which again plays a central role in our approach.

LIMA, Carlos Vinicius Gomes Costa. Sobre b-coloração de grafos com cintura pelo menos 6. 2013. 60 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2013. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-11T12:13:18Z No. of bitstreams: 1 2013_dis_cvgclima.pdf: 3781619 bytes, checksum: 164aea3629d83f1d6d8ba3efcf3ec056 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-14T15:33:44Z (GMT) No. of bitstreams: 1 2013_dis_cvgclima.pdf: 3781619 bytes, checksum: 164aea3629d83f1d6d8ba3efcf3ec056 (MD5) / Made available in DSpace on 2016-07-14T15:33:44Z (GMT). No. of bitstreams: 1 2013_dis_cvgclima.pdf: 3781619 bytes, checksum: 164aea3629d83f1d6d8ba3efcf3ec056 (MD5) Previous issue date: 2013 / The coloring problem is among the most studied in the Graph Theory due to its great theoretical and practical importance. Since the problem of coloring the vertices of a graph G either with the smallest amount of colors is NP-hard, various coloring heuristics are examined to obtain a proper colouring with a reasonably small number of colors. Given a graph G, the b heuristic of colouring comes down to decrease the amount of colors in a proper colouring c, so that, if all vertices of a color class fail to see any color in your neighborhood, then we can change the color to any color these vertices nonexistent in your neighborhood. Thus, we obtain a coloring c ′ with a color unless c. Irving and Molove defined the b-coloring of a graph G as a coloring where every color class has a vertex that is adjacent the other color classes. These vertices are called b-vertices. Irving and Molove also defined the b-chromatic number as the largest integer k, such that G admits a b-coloring by k colors. They showed that determine the value of the b-chromatic number of any graph is NP-hard, but polynomial for trees. Irving and Molove also defined the m-degree of a graph, which is the largest integer m(G) such that there are m(G) vertices with degree at least m(G) − 1. Irving and Molove showed that the m-degree is an upper limit to the b-chromatic number and showed that it is m(T) or m(T)−1 to every tree T, where its value is m(T) if, and only if, T has a good set. In this dissertation, we analyze the relationship between the girth, which is the size of the smallest cycle, and the b-chromatic number of a graph G. More specifically, we try to find the smallest integer g ∗ such that if the girth of G is at least g ∗ , then the b-chromatic number equals m(G) or m(G)−1. Show that the value of g ∗ is at most 6 could be an important step in demonstrating the famous conjecture of Erd˝os-Faber-Lov´asz, but the best known upper limit to g ∗ is 9. We characterize the graphs whose girth is at least 6 and not have a good set and show how b-color them optimally. Furthermore, we show how b-color, also optimally, graphs whose girth is at least 7 and not have good set. / O problema de coloração está entre os mais estudados dentro da Teoria dos Grafos devido a sua grande importância teorica e prática. Dado que o problema de colorir os vértices de um grafo G qualquer com a menor quantidade de cores é NP-difícil, várias heurísticas de coloração são estudadas a fim de obter uma coloração própria com um número de cores razoavelmente pequeno. Dado um grafo G, a heurística b de coloração se resume a diminuir a quantidade de cores utilizadas em uma coloração própria c, de modo que, se todos os vértices de uma classe de cor deixam de ver alguma cor em sua vizinhança, então podemos modificar a cor desses vértices para qualquer cor inexistente em sua vizinhança. Dessa forma, obtemos uma coloração c′ com uma cor a menos que c. Irving e Molove definiram a b-coloração de um grafo G como uma coloração onde toda classe de cor possui um vértice que é adjacente as demais classes de cor. Esses vértices são chamados b-vértices. Irving e Molove também definiram o número b-cromático como o maior inteiro k tal que G admite uma b-coloração por k cores. Eles mostraram que determinar o número b-cromático de um grafo qualquer é um problema NP-difícil, mas polinomial para árvores. Irving e Molove também definiram o m-grau de um grafo, que é o maior inteiro m(G) tal que existem m(G) vértices com grau pelo menos m(G)−1. Irving e Molove mostraram que o m-grau é um limite superior para número b-cromático e mostraram que o mesmo é igual a m(T) ou a m(T)−1, para toda árvore T, onde o número b-cromático é igual a m(T) se, e somente se, T possui um conjunto bom. Nesta dissertação, verificamos a relação entre a cintura, que é o tamanho do menor ciclo, e o número b-cromático de um grafo G. Mais especificamente, tentamos encontrar o menor inteiro g∗ tal que, se a cintura de G é pelo menos g∗, então o número b-cromático é igual a m(G) ou m(G)−1. Mostrar que o valor de g∗ é no máximo 6 poderia ser um passo importante para demonstrar a famosa Conjectura de Erdós-Faber-Lovasz, mas o melhor limite superior conhecido para g∗ é 9. Caracterizamos os grafos cuja cintura é pelo menos 6 e não possuem um conjunto bom e mostramos como b-colori-los de forma ótima. Além disso, mostramos como bicolorir, também de forma ótima, os grafos cuja cintura é pelo menos 7 e não possuem conjunto bom.

Estudamos o problema de famílias intersectantes extremais em um subconjunto aleatório da família dos subconjuntos com exatamente k elementos de um conjunto dado. Obtivemos uma descrição quase completa da evolução do tamanho de tais famílias. Versões semelhantes do problema foram estudadas por Balogh, Bohman e Mubayi em 2009, e por Hamm e Kahn, e Balogh, Das, Delcourt, Liu e Sharifzadeh de maneira concorrente a este trabalho. / We studied the problem of maximal intersecting families in a random subset of the family of subsets with exactly k elements from a given set. We obtained a nearly complete description of the evolution of the size of such families. Similar versions of this problem have been studied by Balogh, Bohman and Mubayi in 2009, and by Hamm and Kahn, and Balogh, Das, Delcourt, Liu and Sharifzadeh concurrently with this work.

Dans le premier chapitre de cette thèse, nous passons en revue les outils de la théorie analytique des nombres qui seront utiles pour la suite. Nous faisons aussi un survol des entiers y−friables, c’est-à-dire des entiers dont chaque facteur premier est plus petit ou égal à y. Au deuxième chapitre, nous présenterons des problèmes classiques de la théorie des nombres probabiliste et donnerons un bref historique d’une classe de fonctions arithmétiques sur un espace probabilisé. Le problème de Erdos sur la table de multiplication demande quel est le nombre d’entiers distincts apparaissant dans la table de multiplication N × N. L’ordre de grandeur de cette quantité a été déterminé par Kevin Ford (2008). Dans le chapitre 3 de cette thèse, nous étudions le nombre d’ensembles y−friables de la table de multiplication N × N. Plus concrètement, nous nous concentrons sur le changement du comportement de la fonction A(x, y) par rapport au domaine de y, où A(x, y) est une fonction qui compte le nombre d’entiers y− friables distincts et inférieurs à x qui peuvent être représentés comme le produit de deux entiers y− friables inférieurs à p x. Dans le quatrième chapitre, nous prouvons un théorème de Erdos-Kac modifié pour l’ensemble des entiers y− friables. Si !(n) est le nombre de facteurs premiers distincts de n, nous prouvons que la distribution de !(n) est gaussienne pour un certain domaine de y en utilisant la méthode des moments. / The object of the first chapter of this thesis is to review the materials and tools in analytic number theory which are used in following chapters. We also give a survey on the development concerning the number of y−smooth integers, which are integers free of prime factors greater than y. In the second chapter, we shall give a brief history about a class of arithmetical functions on a probability space and we discuss on some well-known problems in probabilistic number theory. We present two results in analytic and probabilistic number theory. The Erdos multiplication table problem asks what is the number of distinct integers appearing in the N × N multiplication table. The order of magnitude of this quantity was determined by Kevin Ford (2008). In chapter 3 of this thesis, we study the number of y−smooth entries of the N × N multiplication. More concretely, we focus on the change of behaviour of the function A(x,y) in different ranges of y, where A(x,y) is a function that counts the number of distinct y−smooth integers less than x which can be represented as the product of two y−smooth integers less than p x. In Chapter 4, we prove an Erdos-Kac type of theorem for the set of y−smooth integers. If !(n) is the number of distinct prime factors of n, we prove that the distribution of !(n) is Gaussian for a certain range of y using method of moments.

The following problem has been known for its beauty and elementary character. The Erd˝os Szekeres problem[7]: For any integer k ≥ 3, determine if there exists a smallest positive integer N(k) such that any set of atleast N(k) points in general position in the plane(i.e no three points are in a line) contains k points that are the vertices of a convex k-gon. The finiteness of (k)is proved by Erd˝os and Szekeres using Ramsey theory[7]. In 1978, Erd˝os [6] raised a similar question on empty convex k-gon (convex k-gon without out any interior points) and it has been extensively studied[18]. Several other variants like the convex k-gon with specified number interior points[2] and the chromatic variant[5] have been well studied. In this thesis, we introduce the following two player game variant of the Erd˝os Szekeres problem: Consider a two player game where each player places a point in the plane such that the point set formed will not contain a convex k-gon. The game will end when a convex k-gon is formed and the player who placed the last point loses the game. In our thesis we show a winning strategy forplayer2inthe convex5-gongame and the empty convex5-gongame and argue that the game always ends in the 9th step. We also give an alternative proof for the statement that any point set containing 10 or more points contains an empty convex 5-gon.