This chapter provides an introduction to the study of extremal problems in graph theory, beginning with the classical theorem of Turán. We next turn to bipartite graphs, beginning with trees and paths, and then proving upper bounds for complete bipartite graphs and lower bounds for even cycles. In the process, we take the opportunity to introduce the reader to the Erdős–Rényi random graph G(n, p), which is the central topic of Chapter 5, and also to the fundamental techniques of rotation-extension, double-counting using convexity, and the alteration method, using the inequalities of Markov and Chebyshev. In the second half of the chapter we introduce the notions of supersaturation and stability, which both play key roles in modern research, and prove the Erdős–Stone theorem, often called the fundamental theorem of extremal graph theory, in the case χ(H) = 3.

This chapter gives an overview of several classical topics in the study of graph theory, including perfect matchings, Hamilton cycles, Eulerian trails, proper vertex- and edge-colourings, and connectivity. We begin by proving Hall’s theorem on perfect matchings, Kőnig’s theorem on vertex-covers, and Dirac’s theorem on the minimum degree threshold for a graph to contain a Hamilton cycle. The middle third of the chapter focuses on proper colourings; in particular, we give elegant proofs of Brooks’ theorem on vertex-colourings and Vizing’s theorem on edge-colourings. To finish the chapter, we prove the famous Max-Flow Min-Cut theorem of Ford and Fulkerson, and the fundamental theorems of Menger and Mader on k-connectivity

In this final chapter we study in more detail the properties of the Erdős–Rényi random graph G(n, p). The first half of the chapter introduces the concept of a threshold, covers fundamental results such as the threshold for containing a fixed subgraph and for being connected, and gives Erdős’ classic probabilistic construction of graphs with high girth and chromatic number. The second half of the chapter, which is aimed at Masters students, covers some more advanced material, including the problem of finding spanning subgraphs in G(n, p), the threshold for Hamiltonicity, and the emergence of the giant component. In particular, the final two sections provide striking examples of the power of pseudorandomness.

This chapter introduces the reader to graph Ramsey theory, an area that has seen dramatic progress over the past couple of decades, and in particular over the past few years. We cover the foundational results of several different directions of modern research, including off-diagonal and multicolour Ramsey numbers, Ramsey numbers of sparse graphs, size Ramsey numbers, Ramsey–Turán numbers, and hypergraph Ramsey numbers. In the process, we take the opportunity to introduce the key concept of a pseudorandom graph, and also to see some slightly more advanced applications of ideas (both extremal and probabilistic) that were introduced in Chapter 2.

In this first chapter, we provide a gentle introduction to several basic but fundamental concepts in graph theory, such as paths, trees, connected components, independent sets and the chromatic number, illustrating each definition with some simple but important properties that will be used in later chapters. We also take the opportunity to introduce (in a very simple setting) the probabilistic method, which plays a central role in the book. In the second half of the chapter we introduce the reader to planar graphs, and provide a brief taste of some simple results from the areas of extremal graph theory and Ramsey theory, which are the central topics of Chapters 2 and 4, respectively.

This chapter presents key quantum mechanics principles essential for understanding quantum computation. The postulates of quantum mechanics, mixed states, and density matrices are introduced, along with the Stern–Gerlach experiment’s role in illustrating quantum behavior. Topics such as quantum coherence, entanglement, and the EPR paradox are covered to clarify the fundamental distinctions between classical and quantum systems. Measurement is explored with an emphasis on positive operator-valued measures (POVM), a key concept in understanding quantum state collapse. These principles provide a foundation for studying quantum computation and are essential for understanding qubit behavior, quantum information processing, and subsequent algorithmic structures.

This chapter delves into the quantum circuit model, a primary framework for quantum computation. It begins with the qubit, exploring its representation on the Bloch sphere and its probabilistic measurement outcomes. Quantum gates are introduced as the basic operational units, transforming qubits via unitary operations. The chapter discusses single- and two-qubit gates, building up to universal quantum computation, which enables any quantum function to be constructed through a finite set of gates. This chapter provides an in-depth understanding of information processing in quantum circuits, establishing a practical foundation for executing quantum algorithms and advancing to topics like entanglement-based operations and fault-tolerant design in later chapters.

This chapter introduces seminal quantum algorithms that illustrate quantum computation’s efficiency over classical methods. The Deutsch and Deutsch–Jozsa algorithms showcase quantum parallelism, offering solutions to specific problems with fewer computational steps. The quantum Fourier transform (QFT) is introduced, underpinning period-finding algorithms as well as Shor’s algorithm for integer factorization, which has major implications for cryptography. Grover’s algorithm demonstrates a quadratic speedup for unstructured search problems. By using superposition, entanglement, and phase manipulation, these algorithms highlight the computational power of quantum mechanics and its potential to outperform classical techniques, particularly for complex or classically intractable tasks.