Synchronizing Times for kk-sets in Automata

An automaton is synchronizing if there is a word that maps all states onto the same state. \v{C}ern\'{y}'s conjecture on the length of the shortest such word is probably the most famous open problem in automata theory. We consider the closely related question of determining the minimum length of a word that maps $k$ states onto a single state. For synchronizing automata, we improve the upper bound on the minimum length of a word that sends some triple to a a single state from $0.5n^2$ to $\approx 0.19n^2$. We further extend this to an improved bound on the length of such a word for 4 states and 5 states. In the case of non-synchronizing automata, we give an example to show that the minimum length of a word that sends $k$ states to a single state can be as large as $\Theta\left(n^{k-1}\right)$.Comment: 19 pages, 4 figure

Common Pairs of Graphs

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We extend this notion to an asymmetric setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimized by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. We extend many of the results on common graphs to this asymmetric setting. In addition, we obtain several novel results for common pairs of graphs with no natural analogue in the symmetric setting. We also obtain new examples of common graphs in the classical sense and propose several open problems.Comment: 50 page

Topics in Extremal and Probabilistic Combinatorics.

PhD ThesisThis thesis encompasses several problems in extremal and probabilistic combinatorics. Chapter 1. Tuza's famous conjecture on the saturation number states that for r-uniform hypergraphs F the value sat(F; n)=nr1 converges. I answer a question of Pikhurko concerning the asymptotics of the saturation number for families of hypergraphs, proving in particular that sat(F; n)=nr1 need not converge if F is a family of r-uniform hypergraphs. Chapter 2. Cern y's conjecture on the length of the shortest reset word of a synchronizing automaton is arguably the most long-standing open problem in the theory of nite automata. We consider the minimal length of a word that resets some k-tuple. We prove that for general automata if this is nite then it is nk1 . For synchronizing automata we improve the upper bound on the minimal length of a word that resets some triple. Chapter 3. The existence of perfect 1-factorizations has been studied for various families of graphs, with perhaps the most famous open problem in the area being Kotzig's conjecture which states that even-order complete graphs have a perfect 1-factorization. In my work I focus on another well-studied family of graphs: the hypercubes. I answer almost fully the question of how close (in some particular sense) to perfect a 1-factorization of the hypercube can be. Chapter 4. The k-nearest neighbour random geometric graph model puts vertices randomly in a d-dimensional box and joins each vertex to its k nearest neighbours. I nd signi cantly improved upper and lower bounds on the threshold for connectivity for the k-nearest neighbour graph in high dimensions. ii

Off-Diagonal Commonality of Graphs via Entropy

A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair $(H_1,H_2)$ of such graphs, there exists $p\in(0,1)$ such that an appropriate linear combination of red copies of $H_1$ and blue copies of $H_2$ is minimized by a quasirandom colouring in which $p\binom{n}{2}$ edges are red; such a pair $(H_1,H_2)$ is said to be $(p,1-p)$-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a $(p,1-p)$-common pair $(H_1,H_2)$ such that $H_2$ is uncommon.Comment: 29 pages. Several results and open problems which appear here appeared in early arXiv versions of the paper 'Common Pairs of Graphs' arXiv:2208.02045 which was later split into two papers, of which this is the secon

Improved bounds for cross-Sperner systems

A collection of families (F1,F2,⋯,Fk)∈P([n])k is cross-Sperner if there is no pair i≠j for which some Fi∈Fi is comparable to some Fj∈Fj. Two natural measures of the 'size' of such a family are the sum ∑ki=1|Fi| and the product ∏ki=1|Fi|. We prove new upper and lower bounds on both of these measures for general n and k≥2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011

The rainbow saturation number is linear

Given a graph H , we say that an edge-colored graph G is H -rainbow saturated if it does not contain a rainbow copy of H, but the addition of any nonedge in any color creates a rainbow copy of H. The rainbow saturation number rsat (n,H) is the minimum number of edges among all H-rainbow saturated edge-colored graphs on n vertices. We prove that for any nonempty graph H, the rainbow saturation number is linear in n, thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz

A collection of open problems in celebration of Imre Leader's 60th birthday

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics