Problems in extremal and probabilistic combinatorics

In this thesis we consider some problems in extremal and probabilistic combinatorics.
In Chapter 2 we determine the maximum number of induced cycles that can be contained in a graph on n ≥ n0 vertices, and show that there is a unique graph that achieves this maximum. This answers a question of Tuza.
Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd,Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \ E(G) creates a copy of Qm in G. In Chapter 3, we show that for every fixed m ≥ 2 the minimum number of edges in a (Qd,Qm)-saturated graph is Θ(2d). This answers a question of Johnson and Pinto.
We also answer another question of Johnson and Pinto about weak saturation. Given graphs F and H, a spanning subgraph G of F is said to be weakly (F,H)-saturated if the edges of E(F) \setminus E(G) can be added to G one at a time so that each additional edge creates a new copy of H. We determine the minimum number of edges in a weakly (Qd,Qm)-saturated graph for all d ≥ m ≥ 1. More generally, we determine the minimum number of edges in a subgraph of the d-dimensional grid Pk_d which is weakly saturated with respect to 'axis aligned' copies of a smaller grid Prm.
In Chapter 4 we consider the r-neighbour bootstrap process in the hypercube. This process starts with an initial set A0 of infected vertices in a graph G and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A0 percolates.
We prove a conjecture of Balogh and Bollobás which says that, for fixed r and d tending to infinity, every percolating set in the d-dimensional hypercube has cardinality at least 1+o(1)/r (d choose r-1). We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and weak-saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r=3, we determine the exact cardinality of a minimum percolating set in the d-dimensional hypercube, for all d ≥ 3.
Finally, we consider a more general bootstrap process in a hypergraph setting. Given an r-uniform hypergraph 𝓗, the 𝓗-bootstrap process starts with an initial set of infected vertices of 𝓗 and, at each step, a healthy vertex becomes infected if there exists a hyperedge of 𝓗 in which it is the only healthy vertex. The initial set of infected vertices is said to percolate if every vertex of 𝓗 is eventually infected.
In Chapter 5, for fixed r and large d, we obtain a sharp threshold for the probability that a p-random set of vertices in a q-random subhypergraph of 𝓗 percolates when p,q= Θ(d-1/(r-1)) and 𝓗 is any nearly d-regular r-uniform hypergraph with at most dO(1) vertices which satisfies certain 'codegree' conditions. As it turns out, for this wide class of hypergraphs, the threshold depends only on r and not on the underlying structure of the hypergraph. We apply this result to obtain a sharp threshold for a variant of the graph bootstrap process for strictly 2-balanced graphs. This result generalises a theorem of Korándi, Peled and Sudakov and the proof involves an application of the differential equations method.

Advisor: Scott, A