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It is easy to see that every k-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erdős and Graham asked to estimate the smallest L(k) such that every k-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most L(k). Recently, Girão and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O({2^k}/({k^{1-o(1)}}))$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.
The cry ‘what to do with the Crystal Palace’ continues to reverberate long after the Palace’s fiery demise. Whilst local heritage groups continue to cherish it, its memory has been jeopardised by authorities, both bureaucratic (who have failed to implement a coherent conservation plan for the site) and academic (who have largely refused to engage with building or exhibition). The result, the mental dismantling of the Sydenham Palace from nineteenth-century histories, has been explained by scholarly aversion to reconstruction/inauthenticity and play/populist entertainment, the very aspects which defined it. This chapter explore a small part of the Palace, the Pompeian Court, through our own digital visualisation, housed in Second Life, a popular multi-user online virtual world. By choosing such a venue, we have favoured the pursuit not of absolute authenticity but of virtual presence, offering a space in which visitors to the model, through their avatars, might circulate the space, interact with each other and the exhibits.
The chapter is dedicated to the conceptual artist group ACT. Its historical investigation of the group’s aesthetic strategies attempts to situate them within those structural changes that took place in the aftermath of independence following the collapse of the USSR in 1991 and defined the trajectory for this decade. The chapter investigates, describe and critically revisits the social and cultural context defined as one of a ‘crisis of negation’. Further it analyses those spaces and possibilities that emerge in the gaps between ‘pure creation,’ and are made operational throughout the group’s existence, and the intensity of everyday life in Armenia in the mid-1990s. It argues that the concept of ‘pure creation’ emerges in the clash between autonomous art, and the intensity of turbulent transformations affecting everyday life. It is this clash that transforms the agenda of ‘pure creation’ into a political-artistic program that rhymes with the positivist assumptions of the post-Soviet liberal democratic state.
This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book describes the outcome of needs-assessments undertaken following the Omagh bombing. It explains that the mental health and wider needs arising from loss and trauma must be incorporated as early as possible into the peace-making and peace-building project. The book looks in some detail at the efforts to understand the mental health and related impact of the violence associated with the Troubles in Northern Ireland over the period 1969 to 2015. It considers the Northern Ireland Centre for Trauma and Transformation (NICTT) and also describes developments in therapy, in training and education, and in research and advocacy. The book draws key conclusions about the approaches that could be taken to address mental health and well-being as an essential component of a peace-building project.
This chapter examines Edmund Spenser's involvement in the Irish legal system and suggests how it may have influenced his later conception of Irish law in A view. It assesses the nature of Dublin civic and cultural life and how it would have appeared in the 1580s. The chapter charts Spenser's involvement in a significant debate that took place in Rathfarnham soon after he arrived in Ireland. Many aspects of Dublin life would not have seemed that much different from life in Westminster or Cambridge, especially if Spenser was staying in Dublin Castle as he had lived in Leicester House. As the writings of Bryskett, Stanihurst and others demonstrate, Dublin had no rivals as the cultural and intellectual centre of Ireland, leaving its mark on English settlers such as Edmund Spenser.
This introductory chapter explores and establishes the Sydenham Crystal Palace in relation to existing scholarship on the Great Exhibition of 1851. The Sydenham Palace combined education, entertainment and commerce, and spans both nineteenth and twentieth centuries. We resituate it as an important location within the London art world and establish the broader connections it had with rival ventures such as the South Kensington Museum and the numerous international exhibitions in the period. We set out the new possibilities for the analysis of both nineteenth- and twentieth-century visual and material cultures opened up by this unique venue, problematising the periodisation of art works and attitudes into discretely ‘Victorian’ and ‘Edwardian’ categories.
In 1990 Harold Gauer, former regional director of CARE in the American Midwest, published his professional memoirs entitled Selling Big Charity: The Story of C.A.R.E. This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book focuses on CARE operations overseas, on communications between head and foreign offices, and on the way CARE conducted its business in a foreign environment. The first case study on Korea sheds light on CARE's transition from Europe to "everywhere" and on the way the organization positioned itself in the precarious diplomatic environment of the Cold War. The second case study deals with Egypt. The book analyzes CARE's overseas operations, highlights the reasons for its exponential growth in the 1950s and 1960s. It shows how the new public-private partnership in the field of food relief came about.
We aim to understand how landslides affect the shape and rotational motion of small rubble planetary bodies. We limit ourselves to axisymmetric global landslides and take the primordial shape of the body to be axisymmetric as well. The landslides are modelled as shallow granular surface flows using depth averaging, while incorporating the effects of the body’s rotation, topographical changes from previous landslides, its non-uniform gravity field and possible surface mass shedding. The body’s rotational dynamics is coupled to its shape change due to the transport of regolith – surface grains – and also accounts for the influence of radiation torque. We utilise our framework to investigate regolith motion on idealised rubble bodies and actual asteroids. We then study the evolution of the shape and spin state of an initially spherical rubble asteroid undergoing multiple global landsliding events over millions of years – a time scale comparable to typical asteroidal lifetimes. We find that shape changes due to landsliding resist spin-up due to radiation torque and, in some instances, may even cause the body to spin down. Furthermore, rotational fission is delayed, and may even be suppressed, by regolith redistribution toward the body’s equator. Finally, top-shaped configurations may emerge rapidly, which may explain the prevalence of top-shaped asteroids in near-Earth orbits.
On July 25, 2025, the French Court of Cassation handed down an important decision concerning the existence in international law of possible exceptions to the functional immunity of agents of a foreign state. The Court of Cassation reached the conclusion that the “principle of functional immunity from jurisdiction in criminal matters” recognized for foreign agents acting in the exercise of their functions cannot be invoked in cases of prosecutions for genocide, crimes against humanity, and war crimes. However, the Court’s internationalist approach, which sought to identify the evolution of custom on the issue by examining relevant state practice, remains exclusively Eurocentric, by examining the practice of only five states, all from the same region of the world.
This chapter discusses the nature of the material spectacles connected with mega-events. It begins by recognising and exploring the importance of the 'material spectacle' aspect of mega-events throughout their history over the course of the modernisation process. Mega-events like Olympics and Expos evidently involve performative spectacle of various kinds. The early Expos contributed 'physical legacies' of various kinds to the long-term development of central areas of their host cities, even if only in intermittent and unsystematic ways. The chapter provides some historical information about the relation between, on the one hand, host cities and, on the other hand, mega-events and the material spectacles often associated with them for the 1850-1970 period. The International Olympic Committee (IOC) and Olympic movement began to get interested in the various possible legacies of Olympic Games events, including tangible legacies for the host city, in the 1990s.
This chapter comprises a general introduction to the topic of independents. It begins with a discussion as to the meaning of the concept, and what is understood by an independent for the purposes of this study. The chapter then examines the international and Irish experience of independents and outlines the central premise of this study and its structure, detailing how the question of an independent presence can be explained. The immediate post-World War II period saw the near extinction of the independent parliamentarian. There are two separate points to make about the importance of the party system. The first concerns the general nature of competition in the party system. More specifically, the more open the party system, the greater the presence of independents. The second point about the party system is that in response to its evolution, independents operate in a Downsian or rational, fashion.
Appointment to the Indian Medical Service (IMS) has been construed as instrumental in making individual fortunes or reputations, and in advancing medical professionalisation. The IMS originated in the early seventeenth century as an adjunct to the military and trading ambitions of the East India Company (EIC). This chapter explores three routes of narratives of the IMS. The first is through an analysis of the aggregate practitioner experience that may be gleaned from D.G. Crawford's Roll of the Indian Medical Service. The second strategy is to review the notice given to conditions of IMS in the lay and medical press. The third opportunity is offered by the letters, diaries, and memoirs written by men during or after their service in India. The chapter examines narratives from men in IMS service to reveal the same concerns that troubled their medical counterparts in Britain around professional opportunity, patronage, income, expenditure, and promotion.