## Extremal results for graphs and hypergraphs and other combinatorial problems

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In this dissertation we present several combinatorial results, primarily concerning extremal problems for graphs and hypergraphs, but also covering some additional topics.

In Chapter 2, we consider the following geometric problem of Croft. Let K be a convex body in R^d that contains a copy of another body S in every possible orientation. Is it always possible to continuously move any one copy of S into another, inside K? We prove that the answer is positive if S is a line segment, but, surprisingly, the answer is negative in dimensions at least four for general S.

In Chapter 3, we study the extremal number of tight cycles. Sós and Verstraëte raised the problem of finding the maximum possible size of an n-vertex r-uniform tight-cycle-free hypergraph. When r=2 this is simply n−1, and it was unknown whether the answer is Θ(n^{r−1}) in general. We show that this is not the case for any r≥3 by constructing r-uniform hypergraphs with n vertices and Ω(n^{r−1}logn/loglogn)=ω(n^{r−1}) edges which contain no tight cycles.

In Chapter 4, we study the following saturation question: how small can maximal k-wise intersecting set systems over [n] be? Balogh, Chen, Hendrey, Lund, Luo, Tompkins and Tran resolved this problem for k=3, and for general k showed that the answer is between c_k·2^{n/(k−1)} and d_k·2^{n/⌈k/2⌉}. We prove that their lower bound gives the correct order of magnitude for all k.

In Chapter 5, we prove that for any r, s with r<s, there are n-vertex graphs containing n^{r−o(1)} copies of K_s such that any K_r is contained in at most one K_s. This gives a natural generalisation of the Ruzsa–Szemerédi (6,3)-problem. We also show that there are properly edge-coloured n-vertex graphs with n^{r−1−o(1)} copies of K_r such that no K_r is rainbow, answering a question of Gerbner, Mészáros, Methuku and Palmer about generalised rainbow Turán numbers.

In Chapter 6, we continue the study of the generalised rainbow Turán problem: how many copies of H can a properly edge-coloured graph on n vertices contain if it has no rainbow copy of F? We determine the order of magnitude in essentially all cases when F is a cycle and H is a path or a cycle. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer.

In Chapter 7, we consider the following problem. Let g(n,H) be the smallest k such that we can assign a k-edge-colouring f_v of K_n to each vertex v in K_n with the property that for any copy H_0 of H in K_n, there is some u∈V(H_0) such that H_0 is rainbow in f_u. This function was introduced by Alon and Ben-Eliezer, and we answer several of their questions. In particular, we determine all connected graphs H for which g(n,H)=n^{o(1)}, and show that for all ε>0 there exists r such that g(n,K_r) = Ω(n^{1−ε}). We also prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov about the so-called hypergraph Erdős–Gyárfás function.

In Chapter 8, we study bootstrap percolation for hypergraphs. Consider the process in which, given a fixed r-uniform hypergraph H and starting with a given n-vertex r-uniform hypergraph G, at each step we add to G all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=K_s^{(r)} with s>r≥3, we show that the number of steps of this process can be Θ(n^r). This answers a recent question of Noel and Ranganathan. We also demonstrate that different and interesting maximal running times can occur for other choices of H.

In Chapter 9, we study an extremal problem about permutations. How many random transpositions (meaning that we swap given pairs of elements with given probabilities) do we need to perform on a deck of cards to ‘shuffle’ it? We study several problems on this topic. Among other results, we show that at least 2n−3 such swaps are needed to uniformly shuffle the first two cards of the deck, proving a conjecture of Groenland, Johnston, Radcliffe and Scott.

In Chapter 10, we study the following extremal problem on set systems introduced by Holzman and Körner. We say that a pair (a,b) of families of subsets of an n-element set is cancellative if whenever A,A′∈a and B∈b satisfy A∪B=A′∪B, then A=A′, and whenever A∈a and B,B′∈b satisfy A∪B=A∪B′, then B=B′. Tolhuizen showed that there exist cancellative pairs with |a||b| about 2.25^n, whereas Holzman and Körner proved an upper bound of 2.326^n. We improve the upper bound to about 2.268^n. This result also improved the then best known upper bound for a conjecture of Simonyi about ‘recovering pairs’ (the Boolean case of the ‘sandglass conjecture’), although the upper bound for Simonyi’s problem has since been further improved.

In Chapter 11 we study a continuous version of Sperner’s theorem. Engel, Mitsis, Pelekis and Reiher showed that an antichain in the continuous cube [0,1]^n must have (n−1)-dimensional Hausdorff measure at most n, and they conjectured that this bound can be attained. This was already known for n=2, and we prove this conjecture for all n.

Chapter 12 has similar motivations to the preceding chapter. A subset A of Z^n is called a weak antichain if it does not contain two elements x and y satisfying x_i<y_i for all i. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain A in Z^n, the sum of the sizes of its (n−1)-dimensional projections must be at least as large as its size |A|. They asked what the smallest possible value of the gap between these two quantities is in terms of |A|. We give an explicit weak antichain attaining the minimum for each possible value of |A|.

Finally, in Chapter 13, we study the following problem. Esperet, Gimbel and King introduced the orientation covering number of a graph G as the smallest k with the property that we can choose k orientations of G such that whenever x, y, z are vertices of G with xy,xz∈E(G), then there is a chosen orientation in which both xy and xz are oriented away from x. We prove that the orientation covering number of G is the same as that of K_{χ(G)}, answering a question of Esperet, Gimbel and King. We also determine the orientation covering numbers of complete graphs.