Turán’s Theorem is a cornerstone of extremal graph theory. This theorem and its generalizations are studied in this chapter, including the Erdős–Stone Theorem, Andrásfai–Erdős–Sós Theorem, the notion of stability, chromatic thresholds, and the determination of Turán numbers for color-critical graphs.

A versatile proof approach in extremal set theory is the Linear Algebra method. In order to show an upper bound on the number of sets satisfying certain conditions, we associate a vector to each set and show that the corresponding collection of vectors is linearly independent in an appropriate vector space. Then we use the elementary fact that the maximum size of a linearly independent set is at most the dimension of the space. This chapter explores this method in its myriad forms.

Agent-based modelling (ABM) in social networks offers a powerful framework for simulating individual behaviours and emergent collective patterns within dynamic and interconnected populations. This chapter explores the conceptual foundations, methodological innovations and behavioural implications of ABM in the context of social networks, with a particular focus on the modelling of communication, influence, cooperation and contagion. Drawing from the fields of behavioural science, network theory and computational social science, the chapter presents ABM as a generative tool for understanding how micro-level decision-making rules produce macro-level phenomena. It also critically evaluates the increasing use of AI-driven agents – including large language model (LLM)-powered agents and synthetic personas – in simulating realistic and context-sensitive behaviours within artificial societies. The chapter engages with ethical and methodological challenges, including representation, explainability and the problem of behavioural validity. A worked example is included, illustrating how agent-based simulations can be applied to study misinformation diffusion and norm formation in online social networks. Ultimately, the chapter argues that agent-based modelling in social networks not only advances Behavioural Data Science methodologically, but also fosters new forms of interpretive insight into the dynamics of collective behaviour in an age of digital mediation.

This chapter explores how elements of ‘sense of place’—place dependence, identity, and attachment—are essential for creating liveable, healthy communities. It discusses how community social interaction, cohesion (sense of community), and collective action enhance liveability. Addressing frequent shortcomings in defining and measuring ‘liveability’, the authors integrate recent geospatial population health research to establish connections between place, health, development, and policy. A framework linking community cognitions, attachment, and social action in response to environmental threats and disruptions is applied to brief case studies from Bulgaria and the United States illustrating varied community responses aimed at healthier, more liveable environments. Recommendations include leveraging liveability audits to evaluate local and regional impacts of disruptions, articulating threats to community identity and economic wellbeing, and embedding psychological conceptions of place into public-health and urban-planning assessments. This integrated, ecological approach aims to strengthen community responses and inform policy decisions supporting population health and wellbeing

How is people’s happiness determined by economic factors such as their income? Big data (particularly, behavioural data at scale) are essential to answering this question, but there is disagreement about the strength of evidence for causal relationships that is given by different types of analysis. This chapter reviews the different approaches to analysis that have been taken. First, it is argued that most existing literature both under-claims regarding the evidence for causality given by some types of analysis of big data, such as correlational analyses, and over-claims for other types of analyses, such as those involving panel data. Thus, even correlational data can be informative to the extent that associations are generally rare and that theoretical targets and alternatives are fully specified and given prior probabilities. Second, a new methodological problem is identified for a specific model of the income–rank relationship. According to the income rank hypothesis, people’s well-being is determined not by their income but by the ranked position that their income occupies within a social comparison group. It is shown by simulation that spurious rank effects can occur in regression analyses if there is noise in measured income, but that this problem can be reduced with the use of robust regression techniques. A new analysis of a large dataset, the Panel Study of Income Dynamics, is reported. The results show that income rank effects are not reduced by the use of robust regression techniques, suggesting that previous support for the income rank hypothesis is not due to an artefact.

Providing a cohesive reference for advanced undergraduates, graduate students, and even experienced researchers, this text contains both introductory and advanced material in extremal graph theory, hypergraph theory, and Ramsey theory. Along the way, this book includes many modern proof techniques in the field, such as the probabilistic method and algebraic methods. Several recent breakthroughs are presented with complete proofs, for example, recent results on the sunflower problem, and off-diagonal and geometric Ramsey theory. It is perhaps unique in containing material on both hypergraph regularity and containers.

Featuring an extensive list of exercises, this book serves as a valuable teaching resource for a variety of courses in extremal combinatorics. Each of the two parts can form the basis of separate courses, and the majority of sections are designed to match the length of a single lecture.

This chapter critically examines ‘sense of place’ within a post-colonial context, addressing its theoretical complexity and the absence of Indigenous perspectives in mainstream discourse. Building on Erfani’s synthesis of over 2,000 studies, it explores four core components—place attachment, identity, satisfaction, and dependence—while highlighting their intersections with psychological sense of community. The chapter emphasises the need to integrate Indigenous worldviews, which conceive humans as inseparable from land and place, contrasting with settler-colonial paradigms rooted in ownership and separation. It advocates for Indigenous-led methodologies to measure sense of place, showcasing frameworks like the Mauri Model and Caring for Country. By bridging gaps between Western and Indigenous perspectives through participatory, Indigenous-led approaches such as ‘working two-ways’ or ‘two-eyed seeing’, the chapter aims to foster mutual understanding and sustainable practices. Ultimately, it calls for a paradigm shift toward holistic, relational conceptions of place to address environmental and cultural challenges in the Anthropocene.

The recent surge in conversational AI has opened up new avenues for the application of Behavioural Data Science, with Generative AI, particularly LLM models such as ChatGPT, representing a promising platform for analysing human behaviour. This chapter provides an overview of the role of Generative AI models in Behavioural Data Science, highlighting their strengths and limitations. The chapter begins by introducing the principles of Generative AI models and their potential applications in Behavioural Data Science. It discusses the advantages of using Generative AI models to study human behaviour, such as their ability to analyse large volumes of unstructured data and their capacity to learn from interactions with users. The chapter then describes the different ways in which Generative AI models can be used in Behavioural Data Science, such as analysing sentiment, predicting behaviour and generating insights from user interactions. It discusses the challenges associated with using Generative AI models, such as the need for accurate training data and the potential for bias in the model. The chapter also addresses the ethical concerns associated with the use of Generative AI models, such as privacy violations and the potential for unintended consequences. It discusses ways to address these concerns, such as implementing transparency and explainability in Generative AI models. The chapter concludes by discussing the future of Generative AI models in Behavioural Data Science, highlighting the potential for interdisciplinary collaboration with other fields such as psychology and sociology. It emphasises the need for continued development and refinement of Generative AI models and their associated methods for studying human behaviour. This chapter provides an overview of the role of Generative AI models in Behavioural Data Science, highlighting their strengths and limitations. It serves as a valuable resource for researchers and practitioners in the field who are interested in utilising Generative AI models to analyse and understand human behaviour.

The success of the linear algebra method in combinatorics has led to other advanced algebraic techniques, often using polynomials in novel ways. This chapter focuses on these developments. Two highlights include the resolution of the Kakeya problem in finite fields and the combinatorial Nullstellensatz of Alon.

This chapter approaches fashion as a technology of transition that modifies bodies and generates gender and sex – in the early modern period and today. It offers a broad and exploratory examination of trans fashion and fashioning in the early modern period as it intersects with premodern trans studies, suggesting “trans fashioning” as one method of understanding not only trans genders but also the construction and maintenance (in part through fashion) of all genders, especially the artifice and impossibility of stable cisgender sex. Read in a trans way, histories of fashion show the patterns and seams and stitches of styling sex and normalizing the fashioning of cis or binary gender. At once superficial and significant, changeable and permanent, the gendering technologies of early modern fashion undercut false modern binaries between “social” and “medical” transition, between what we call sex and gender or cis and trans, and between private and public bodies.

We study extremal numbers for even cycles in this chapter, including constructions from generalized polygons, walk-counting methods including the Blakley–Roy Inequality, tensor trick, and Alon–Hoory–Linial entropy approach for non-backtracking walks, Moore graphs and eigenvalues, the even cycle Theorem of Bondy and Simonovits, and Sidorenko’s Conjecture.

A theory of some finding or observation is an explanation of that finding or observation. Further, a good theory is a set of principles that are sufficient to show that the phenomenon is an instance of more general phenomena or principles. But not all explanations help us understand general phenomena because they lack some fundamental characteristics. The necessary characteristics of adequate explanations include explicit definitions and precise and limited scope, that is, they do not attempt to explain everything about a given event or action. Further, they can be tested with empirical data; they do not appeal to supernatural forces or to explanations with claims that testing is not necessary.

Diasporic communities played a crucial role in the vibrant trading system that flourished across maritime Asia during the Song period. They served as cultural middlemen with both local merchants and the maritime trade offices that supervised the trade, and they provided hospitality for arriving merchants while engaging in their own long-distance business. Within the port cities of Guangzhou and Quanzhou in particular, there were merchant communities from India and Southeast Asia, but by far the largest and best documented were the Muslim communities, which is the focus of this chapter. It covers where and how they lived (and were buried), their relations with the Song authorities, and the elevation of some to official status. It also contrasts their remarkably peaceful history under the Song to the violence that they endured at time during both the Tang and Yuan.

This chapter focuses on Spain, and investigates the culmination of the genre of the printed newspaper in the final years of the seventeenth century by focusing on the so-called Darien Scheme, the attempt by Scottish merchants to create a settlement colony on the Isthmus of Panama in 1699–1700. The Darien Scheme brings the book full circle to the same spot in the Caribbean where newspaper interest had peaked during Drake’s final campaign a century before. Transnational coverage of the Darien Scheme, and its collapse, provides an excellent opportunity to compare the different media landscapes – from strongly engaged (Scotland, England, and Spain) to neutral or indifferent (Dutch Republic, Germany, France, and Italy) – at a time when the European press began to lose its monopoly on periodical news in the Atlantic world.

I present historical background, contemporary status, and potential future development of the psychology of religion (PoR) in Switzerland, beginning with its origins: the formative people, places, and various intellectual schools of thought. Writing about the current state of the topic, I reflect on influential factors that are either facilitating or inhibiting the study of PoR, including publication options, topical emphases, practitioners, orientations, methodologies, and professional organizations.

I offer opinions concerning future development topics that are emerging as important in the immediate future and/or are perennially important in order to stimulate creative and useful research including Western theoretical relevance, the extent to which Western PoR theories may or may not contain reasonable expectations and concepts for this region, contextual nuances, Indigenous theoretical concerns, collaborative research opportunities, and common faux pas – reflections on what people unfamiliar with this region commonly and incorrectly assume about conducting PoR work in this context.

Individuals have a surprisingly high capacity for making decisions quickly and still considering a multitude of information. This capability – often referred to as intuition – relies on automatic processes that can be described with neural networks. Particularly parallel constraint satisfaction (PCS) networks – a specific type of interactive activation networks – have been successful in capturing multiple aspects of choice behaviour. PCS models include restrictions to neural networks that capture specific features of cognition. This chapter will describe how PCS and other content models of decision-making can be evaluated and potentially improved by using artificial intelligence, specifically generic multi-layer (deep learning) neural network models. It will exemplify how choice behaviour can be modelled and predicted with PCS. The predictive performance of PCS will be contrasted with that of a generic neural network model. Possibilities and implications for the improvement of content models for choice behaviour using artificial intelligence are discussed.

Misogynist attacks on women for a purportedly vain and frivolous preoccupation with fashion have proliferated for centuries. This chapter focuses on a feminist response to such rhetoric by a seventeenth century Italian nun, Arcangela Tarabotti. Forced to enter a convent as a teenager, Tarabotti defended women's free enjoyment of fashion and pointed out men’s obsession with adornment. As Eugenia Paulicelli shows, Tarabotti’s theory of fashion offers a feminist critique of masculinity and patriarchy, links women’s right to fashion to their right to education, and articulates the value of women’s work with texts and textiles. Through her engagement with fashion, Tarabotti recasts early modern ideas about gender, as well as distinctions between bodily and intellectual pursuits. She also cannily takes advantage of the ways that her own writing and nuns’ traditional needlework are able to circulate despite their authors’ physical confinement.