Extremal combinatorics, graph limits and computational complexity

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0, 1} where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the ‘saturation number’ of Qm in Qd. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Qd of bounded average degree such that G does not contain a copy of Qm but, for every G′ such that G ( G′ ⊆ Qd, the graph G′ contains a copy of Qm. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2(1−ε)k subsets of [n] which does not contain a chain of length k + 1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0, 1) such that the smallest such collection is of cardinality 2 for all k. In Chapter 4, we obtain an exact expression for the ‘weak saturation number’ of Qm in Qd. That is, we determine the minimum number of edges in a spanning subgraph G of Qd such that the edges of E(Qd)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Qm. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of ‘axis aligned’ copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A0 of ‘infected’ vertices in a graph G and, at each step, a ‘healthy’ vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A0 percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Qd has cardinality at least 1+o(1) r ( d r−1 ) . This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r = 3 (the minimum cardinality in the case r = 2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset V (q, n) consisting of all subspaces of Fq ordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed ‘container method,’ to obtain an upper bound on the number of antichains in V (q, n) and a bound on the size of the largest antichain in a p-random subset of V (q, n) which holds with high probability for p in a certain range. In Chapter 8, we construct a ‘finitely forcible graphon’ W for which there exists a sequence (εi)i=1 tending to zero such that, for all i ≥ 1, every weak εi-regular partition ofW has at least exp ( ε−2 i /2 5 log∗ ε−2 i ) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p, q with p/q ≥ 2, a circular (p, q)-colouring of a graph G is a mapping V (G) → Zp such that any two adjacent vertices are mapped to elements of Zp at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p, q)colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a (p, q)colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4. Statement of Originality This thesis contains no material which has been accepted in whole, or in part, for any other degree or diploma. Chapters 2, 3 and 4 are based on two joint papers with Natasha Morrison and Alex Scott [170,171]. Chapter 5 is based on a joint paper with Natasha Morrison [169]. Chapters 6 and 7 are based on a joint paper with Alex Scott and Benny Sudakov [176]. Chapter 8 is based on a joint paper with Jacob Cooper, Tomáš Kaiser and Daniel Král’ [74]. Chapter 9 is based on a joint paper with Richard Brewster, Sean McGuinness and Benjamin Moore [55].