Resilient and exposure-resilient functions are functions whose output appears random even if some portion of their input is either revealed or fixed. We explore an alternative way of characterizing these objects that ties them explicitly to the theory of randomness extractors and simplifies current proofs of basic results. We also describe the inclusions and separations governing the various classes of resilient and exposure-resilient functions. Using this knowledge, we explore the possibility of improving existing constructions of these functions and prove that one specific method of doing so is impossible.

Dois problemas combinatórios são estudados: (i) determinar a quantidade de cópias de um hipergrafo fixo em um hipergrafo uniforme pseudoaleatório, e (ii) estimar números de Ramsey de ordem dois e três para grafos com largura de banda pequena e grau máximo limitado. Apresentamos um lema de contagem para estimar a quantidade de cópias de um hipergrafo k-uniforme linear livre de conectores (conector é uma generalização de triângulo, para hipergrafos) que estão presentes em um hipergrafo esparso pseudoaleatório G. Considere um hipergrafo k-uniforme linear H que é livre de conectores e um hipergrafo k-uniforme G com n vértices. Seja d_H=\\max\\{\\delta(J): J\\subset H\\} e D_H=\\min\\{k d_H,\\Delta(H)\\}. Estabelecemos que, se os vértices de G não possuem grau grande, famílias pequenas de conjuntos de k-1 elementos de V(G) não possuem vizinhança comum grande, e a maioria dos pares de conjuntos em {V(G)\\choose k-1} possuem a quantidade ``correta\'\' de vizinhos, então a quantidade de imersões de H em G é (1+o(1))n^{|V(H)|}p^{|E(H)|}, desde que p\\gg n^{1/D_H} e |E(G)|={n\\choose k}p. Isso generaliza um resultado de Kohayakawa, Rödl e Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], que provaram que, para p dado como acima, esse lema de imersão vale para grafos, onde H é um grafo livre de triângulos. Determinamos assintoticamente os números de Ramsey de ordem dois e três para grafos bipartidos com largura de banda pequena e grau máximo limitado. Mais especificamente, determinamos assintoticamente o número de Ramsey de ordem dois para grafos bipartidos com largura de banda pequena e grau máximo limitado, e o número de Ramsey de ordem três para tais grafos, com a suposição adicional de que ambas as classes do grafo bipartido têm aproximadamente o mesmo tamanho. / Two combinatorial problems are studied: (i) determining the number of copies of a fixed hipergraph in uniform pseudorandom hypergraphs, and (ii) estimating the two and three color Ramsey numbers for graphs with small bandwidth and bounded maximum degree. We give a counting lemma for the number of copies of linear k-uniform \\emph hypergraphs (connector is a generalization of triangle for hypergraphs) that are contained in some sparse hypergraphs G. Let H be a linear k-uniform connector-free hypergraph and let G be a k-uniform hypergraph with n vertices. Set d_H=\\max\\{\\delta(J)\\colon J\\subset H\\} and D_H=\\min\\{kd_H,\\Delta(H)\\}. We proved that if the vertices of G do not have large degree, small families of (k-1)-element sets of V(G) do not have large common neighbourhood and most of the pairs of sets in {V(G)\\choose k-1} have the `right\' number of common neighbours, then the number of embeddings of H in G is (1+o(1))n^p^, given that p\\gg n^ and |E(G)|=p. This generalizes a result by Kohayakawa, R\\\"odl and Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], who proved that, for p as above, this result holds for graphs, where H is a triangle-free graph. We determine asymptotically the two and three Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that both classes of the bipartite graph have almost the same size.

Dois problemas combinatórios são estudados: (i) determinar a quantidade de cópias de um hipergrafo fixo em um hipergrafo uniforme pseudoaleatório, e (ii) estimar números de Ramsey de ordem dois e três para grafos com largura de banda pequena e grau máximo limitado. Apresentamos um lema de contagem para estimar a quantidade de cópias de um hipergrafo k-uniforme linear livre de conectores (conector é uma generalização de triângulo, para hipergrafos) que estão presentes em um hipergrafo esparso pseudoaleatório G. Considere um hipergrafo k-uniforme linear H que é livre de conectores e um hipergrafo k-uniforme G com n vértices. Seja d_H=\\max\\{\\delta(J): J\\subset H\\} e D_H=\\min\\{k d_H,\\Delta(H)\\}. Estabelecemos que, se os vértices de G não possuem grau grande, famílias pequenas de conjuntos de k-1 elementos de V(G) não possuem vizinhança comum grande, e a maioria dos pares de conjuntos em {V(G)\\choose k-1} possuem a quantidade ``correta\'\' de vizinhos, então a quantidade de imersões de H em G é (1+o(1))n^{|V(H)|}p^{|E(H)|}, desde que p\\gg n^{1/D_H} e |E(G)|={n\\choose k}p. Isso generaliza um resultado de Kohayakawa, Rödl e Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], que provaram que, para p dado como acima, esse lema de imersão vale para grafos, onde H é um grafo livre de triângulos. Determinamos assintoticamente os números de Ramsey de ordem dois e três para grafos bipartidos com largura de banda pequena e grau máximo limitado. Mais especificamente, determinamos assintoticamente o número de Ramsey de ordem dois para grafos bipartidos com largura de banda pequena e grau máximo limitado, e o número de Ramsey de ordem três para tais grafos, com a suposição adicional de que ambas as classes do grafo bipartido têm aproximadamente o mesmo tamanho. / Two combinatorial problems are studied: (i) determining the number of copies of a fixed hipergraph in uniform pseudorandom hypergraphs, and (ii) estimating the two and three color Ramsey numbers for graphs with small bandwidth and bounded maximum degree. We give a counting lemma for the number of copies of linear k-uniform \\emph hypergraphs (connector is a generalization of triangle for hypergraphs) that are contained in some sparse hypergraphs G. Let H be a linear k-uniform connector-free hypergraph and let G be a k-uniform hypergraph with n vertices. Set d_H=\\max\\{\\delta(J)\\colon J\\subset H\\} and D_H=\\min\\{kd_H,\\Delta(H)\\}. We proved that if the vertices of G do not have large degree, small families of (k-1)-element sets of V(G) do not have large common neighbourhood and most of the pairs of sets in {V(G)\\choose k-1} have the `right\' number of common neighbours, then the number of embeddings of H in G is (1+o(1))n^p^, given that p\\gg n^ and |E(G)|=p. This generalizes a result by Kohayakawa, R\\\"odl and Sissokho [Embedding graphs with bounded degree in sparse pseudo\\-random graphs, Israel J. Math. 139 (2004), 93--137], who proved that, for p as above, this result holds for graphs, where H is a triangle-free graph. We determine asymptotically the two and three Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that both classes of the bipartite graph have almost the same size.

This thesis focuses on applications of classical tools from probability theory and convex analysis such as limit theorems to problems in theoretical computer science, specifically to pseudorandomness and learning theory. At first look, limit theorems, pseudorandomness and learning theory appear to be disparate subjects. However, as it has now become apparent, there's a strong connection between these questions through a third more abstract question: what do random objects look like. This connection is best illustrated by the study of the spectrum of Boolean functions which directly or indirectly played an important role in a plethora of results in complexity theory. The current thesis aims to take this program further by drawing on a variety of fundamental tools, both classical and new, in probability theory and analytic geometry. Our research contributions broadly fall into three categories. Probability Theory: The central limit theorem is one of the most important results in all of probability and richly studied topic. Motivated by questions in pseudorandomness and learning theory we obtain two new limit theorems or invariance principles. The proofs of these new results in probability, of interest on their own, have a computer science flavor and fall under the niche category of techniques from theoretical computer science with applications in pure mathematics. Pseudorandomness: Derandomizing natural complexity classes is a fundamental problem in complexity theory, with several applications outside complexity theory. Our work addresses such derandomization questions for natural and basic geometric concept classes such as halfspaces, polynomial threshold functions (PTFs) and polytopes. We develop a reasonably generic framework for obtaining pseudorandom generators (PRGs) from invariance principles and suitably apply the framework to old and new invariance principles to obtain the best known PRGs for these complexity classes. Learning Theory: Learning theory aims to understand what functions can be learned efficiently from examples. As developed in the seminal work of Linial, Mansour and Nisan (1994) and strengthened by several follow-up works, we now know strong connections between learning a class of functions and how sensitive to noise, as quantified by average sensitivity and noise sensitivity, the functions are. Besides their applications in learning, bounding the average and noise sensitivity has applications in hardness of approximation, voting theory, quantum computing and more. Here we address the question of bounding the sensitivity of polynomial threshold functions and intersections of halfspaces and obtain the best known results for these concept classes.

Recentemente, Barak et al. (2004) exibiram construções de extratores e dispersores determinísticos (funções computáveis em tempo polinomial) com parâmetros melhores do que era anteriormente possível. Introduziremos os conceitos envolvidos em tal trabalho e mencionaremos suas aplicações; em particular, veremos como é possível obter cotas muito melhores para o problema Ramsey bipartido (um problema bem difícil) utilizando as construções descritas no artigo. Também apresentamos resultados originais para melhorar tais construções. Tais idéias são inspiradas no trabalho de Anup Rao (2005) e utilizam o recente êxito de Jean Bourgain (2005) em obter extratores que quebram a \"barreira 1/2\". / Recently, Barak et al. (2004) constructed explicit deterministic extractors and dispersers (these are polynomial-time computable functions) with much better parameters than what was known before. We introduce the concepts involved in such a construction and mention some of its applications; in particular, we describe how it is possible to obtain much better bounds for the bipartite Ramsey problem (a very hard problem) using the machinery developed in that paper. We also present some original results that improve on these constructions. They are inspired by the work of Anup Rao (2005) and uses the recent breakthrough of Jean Bourgain (2005) in obtaining 2-source extractors that break the \"1/2-barrier\".

Recentemente, Barak et al. (2004) exibiram construções de extratores e dispersores determinísticos (funções computáveis em tempo polinomial) com parâmetros melhores do que era anteriormente possível. Introduziremos os conceitos envolvidos em tal trabalho e mencionaremos suas aplicações; em particular, veremos como é possível obter cotas muito melhores para o problema Ramsey bipartido (um problema bem difícil) utilizando as construções descritas no artigo. Também apresentamos resultados originais para melhorar tais construções. Tais idéias são inspiradas no trabalho de Anup Rao (2005) e utilizam o recente êxito de Jean Bourgain (2005) em obter extratores que quebram a \"barreira 1/2\". / Recently, Barak et al. (2004) constructed explicit deterministic extractors and dispersers (these are polynomial-time computable functions) with much better parameters than what was known before. We introduce the concepts involved in such a construction and mention some of its applications; in particular, we describe how it is possible to obtain much better bounds for the bipartite Ramsey problem (a very hard problem) using the machinery developed in that paper. We also present some original results that improve on these constructions. They are inspired by the work of Anup Rao (2005) and uses the recent breakthrough of Jean Bourgain (2005) in obtaining 2-source extractors that break the \"1/2-barrier\".