Topics in Extremal and Probabilistic Combinatorics

This thesis describes my contributions in three distinct areas of extremal and probabilistic combinatorics by solving an open problem in each of these areas. Chapter 2 concerns graph decompositions. It has my solution (obtained with co-authors) of the ascending subgraph decomposition conjecture, a 40-year old conjecture due to Alavi, Boals, Chartrand, Erdős, and Oellerman. Chapter 3 concerns finding spanning subgraphs in the perturbed graph, which is the union of a deterministic and a random graph. It has my work (obtained with co-authors) that a perturbed graph on n vertices whose edges are uniformly coloured with n colours contains, with high probability, a rainbow Hamilton cycle. Chapter 4 concerns random Ramsey theory, where I worked on the two-round random Ramsey game. Here we want to colour two independent copies of the random graph on the same vertex set in two rounds, so that the colouring from the first random graph extends to the second. I proved a lower bound on the number of edges the second random graph can have so that we can find a colouring of the first random graph that does extend, with high probability, to the second one.