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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 ≥ n_{0} vertices, and show that there is a unique graph that achieves this maximum. This answers a question of Tuza. Let Q_{d} denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Q_{d} is said to be (Q_{d},Q_{m})saturated if it does not contain Q_{m} as a subgraph but adding any edge of E(Q_{d}) \ E(G) creates a copy of Q_{m} in G. In Chapter 3, we show that for every fixed m ≥ 2 the minimum number of edges in a (Q_{d},Q_{m})saturated graph is Θ(2^{d}). 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 (Q_{d},Q_{m})saturated graph for all d ≥ m ≥ 1. More generally, we determine the minimum number of edges in a subgraph of the ddimensional grid P_{k}_^{d} which is weakly saturated with respect to 'axis aligned' copies of a smaller grid P_{r}^{m}. In Chapter 4 we consider the rneighbour bootstrap process in the hypercube. This process starts with an initial set A_{0} 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 A_{0} 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 ddimensional hypercube has cardinality at least ^{1+o(1)}/_{r} (d choose r1). We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and weaksaturation. 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 ddimensional hypercube, for all d ≥ 3. Finally, we consider a more general bootstrap process in a hypergraph setting. Given an runiform hypergraph H, the Hbootstrap 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 prandom set of vertices in a qrandom subhypergraph of H percolates when p,q= Θ(d^{1/(r1)}) and H is any nearly dregular runiform hypergraph with at most d^{O(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.
