Defesa de Tese.

Banca examinadora: Robert Morris (IMPA, orientador) Roberto Imbuzeiro Oliveira (IMPA) Maurício Collares (UFMG) Taísa Martins (UFF) Guilherme Oliveira Mota (USP) Suplente: Simon Griffiths (PUC-Rio)

Abstract: In this thesis we study two of the main objects in probabilistic combinatorics: random matrices and random graphs. In the first part, joint with Campos, Morris and Morrison, we consider a uniformly-chosen random symmetric matrix with entries in {-1,+1}. We obtain an ‘exponential-type’ bound on the probability that this matrix is singular. Our main new ingredient is an inverse Littlewood--Offord theorem whose statement is inspired by the method of hypergraph containers. In the second part, joint with Griffiths and Morris, we study the size of the maximum k-clique packing in the random graph G(n,p). A clique packing is just a set of edge-disjoint cliques. For every value of k which is close to the size of the largest clique in G(n,p), we obtain the order of the maximum k-clique packing in G(n,p). To show this result, we follow a random greedy process and use the differential equation method. In the third part, joint with Liebenau, Mendonça and Skokan, we study asymmetric Ramsey properties of G(n,p) for cliques and cycles. For any pair of r-clique and k-cycle, we determine the threshold for finding a red copy of a r-clique or a blue copy of a k-cycle in every red and blue edge-colouring of G(n,p). The main tool behind the proof is a structural characterisation of Ramsey graphs for each pair of r-clique and k-cycle .