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

Sums of Vectors and Turán's Problem for 3-graphs

Let X be a Hilbert space, 0 ⩽ m ⩽ n integers. If a1, ..., an ∈ X, ∥ai∥ ⩾ 1(1 ⩽ i ⩽ n) then N3(a1, ..., an) denotes the number of sums ∥ai1 + ai2 + ai3∥ ⩾ 1. The asymptotic behaviour of N3 (X, n) = min N3(a1, ..., an) is studied