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In this thesis, we present a number of results, mostly concerning set systems that are antichains and/or have bounded diameter. Chapter 1 gives a more detailed outline of the thesis. In Chapter 2, we give a new short proof of Kleitman's theorem concerning the maximal size of a set system with bounded diameter. In Chapter 3, we turn our attention to antichains with bounded diameter. Šileikis conjectured that an antichain of diameter D has size at most (n/D/2). We present several partial results towards the conjecture.
In 2014, Leader and Long gave asymptotic bounds on the size of a set system where |A\B| ≠ 1 and more generally, when |A\B| ≠ k. In Chapter 4, we present streamlined versions of their proofs, with slightly better bounds.
The final chapter presents a proof for the following poset analog of an elementary graph theory problem: every poset with |R| relations contains a height two subposet with at least |R|/2 relations.