Substructure Densities in Extremal Combinatorics

One of the primary goals of combinatorial mathematics is to understand how an object's properties are influenced by the presence or multiplicity of a given substructure. Over time, it has become popular to highlight the asymptotic behaviour of objects by expressing results in terms of the density of substructures. In this thesis, we investigate three topics concerning combinatorial density: We study the interplay between the densities of cycles of length 3 and 4 in large tournaments, we characterise quasirandomness in permutations, and we solve two open problems about the inducibility of trees.