UC San Diego

In this dissertation, we examine various problems in extremal set theory, which typically entails maximizing the size of a collection of subsets, or set family, given intersection constraints. For instance, the classical Erd{\H o}s-Ko-Rado theorem (1961) establishes the largest set family of size $k$ subsets of an $n$ element set which is {\it intersecting} (i.e. has the property that any two sets have at least one element in common). The largest such intersecting family is the collection of all size $k$ subsets that contain a fixed element, which is commonly referred to as {\it the star}. Answering a question of Erd{\H o}s, Ko and Rado, Hilton and Milner (1967) determined the largest intersecting set family which is not isomorphic to a sub-collection of the star. This dissertation settles a conjecture of Hilton and Milner (1967) on the largest set family, for each integer $d \geq 3$, of size $k$ subsets of an $n$ element set which has the property that any $d$ sets have at least one element in common and is not isomorphic to a sub-collection of the star. We also consider various combinatorial results on set families with restricted intersection properties. In particular, we prove generalizations of both the {\it Bollob{ a}s two family} theorem and the {\it Oddtown and Eventown} theorems.