Partition problems in discrete geometry

This thesis deals with the following type of problems, which we denote partition problems, Given a set X in R^d, is there a way to partition X such that the convex hulls of all parts satisfy certain combinatorial properties? We focus on the following two kinds of partition problems. Tverberg type partitions. In this setting, one of the properties we ask the sets to satisfy is that their convex hulls all intersect. Ham sandwich type partitions. In this setting, one of the properties we ask the sets to satisfy is that the interior of their convex hulls are pairwise disjoint. The names for these types of partitions come from the quintessential theorem from each type, namely Tverberg's theorem and the ham sandwich theorem. We present a generalisation and a variation of each of these classic results. The generalisation of the ham sandwich theorem extends the classic result to partitions into any arbitrary number of parts. This is presented in chapter 2. Then, in chapter 3, variations of the ham sandwich theorem are studied when we search for partitions such that every hyperplane avoids an arbitrary number of sections. The generalisation of Tverberg's theorem consists of adding a condition of tolerance to the partition. Namely, that we may remove an arbitrary number of points and the partition still is Tverberg type. This is presented in chapter 4. Then, in chapter 5, ``colourful'' variations of Tverberg's theorem are studied along their applications to some purely combinatorial problems

Extremal Problems on the Hypercube

PhDThe hypercube, Qd, is a natural and much studied combinatorial object, and we discuss various extremal problems related to it. A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number, that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that sat(Qd;Q2) = �� 1 4 + o(1) d2d��1. We also prove upper bounds on sat(Qd;Qm) for general m.Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured a lower bound on the number of edge-disjoint paths between A and B in the directed hypercube. Using an unusual form of the compression argument, we confirm the conjecture by reducing the problem to a the case of the undirected hypercube. We also prove an analogous conjecture for vertex-disjoint paths using the same techniques, and extend both results to the grid. Additionally, we deal with subcube intersection graphs, answering a question of Johnson and Markström of the least r = r(n) for which all graphs on n vertices may be represented as subcube intersection graph where each subcube has dimension exactly r. We also contribute to the related area of biclique covers and partitions, and study relationships between various parameters linked to such covers and partitions. Finally, we study topological properties of uniformly random simplicial complexes, employing a characterisation due to Korshunov of almost all down-sets in the hypercube as a key tool