Introduction: Diets are currently unsustainable in many countries as evidenced by the growing burden of malnutrition, degradation of natural resources, contributions to climate change, and unaffordability of healthy diets. Agreement on what constitutes a healthy and sustainable diet has been debated. In 2019, FAO and WHO published the Sustainable Healthy Diets Guiding Principles, defining what qualifies as a sustainable healthy diet. While valuable, these principles require measurable indicators to support their operationalization. Our scoping review aims to describe how sustainable healthy diets have been assessed in the literature since 2010.Methods: A search for English-language articles published in peer-reviewed journals was conducted from January 2010 through February 2020 across three databases. Out of the 504 articles initially identified, 103 articles were included. Metadata were extracted from each article on: publication year, country of study, study aims, methods, main data sources, indicators used to assess sustainable healthy diets, reported indicator strengths or limitations, and main study findings. A qualitative content analysis identified major conceptual themes across indicators and their frequency of use.Findings: From the 103 empirical articles included in our review, 57.3% were published after 2017. Most studies were carried out in high-income countries (74%). Approximately 42% of the articles assessed the sustainability of diets using solely health and environmental indicators; <25% assessed the sustainability of diets across health, environmental, and sociocultural aspects of sustainability. We found a substantial number of unique indicators used for assessing health (n = 82), environmental (n = 117), and sociocultural (n = 43) aspects of diets. These indicators covered concepts related to health outcomes, aspects of diet quality, natural resources, climate change, cultural acceptability, and cost of diets. The preponderance of indicators currently used in research likely poses challenges for stakeholders to identify the most appropriate measures.Conclusion: Robust indicators for sustainable healthy diets are critical for understanding trends, setting targets, and monitoring progress across national and sub-national levels. Our review highlights the geographical imbalance, the narrow focus on health and environmental aspects, and the lack of common measures used in research. Measures registries could provide the decision-support needed by stakeholders to aid in the indicator selection process.

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Given a 3-uniform hypergraph H consisting of a set V of vertices, and T ⊆ V 3 triples, a null labelling is an assignment of ±1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and −1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.

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We investigate strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors, where vertices represent players and the edges indicate their relations. Each player v chooses one of the available colors as its own strategy, and its payoff (or utility) is the number of neighbors of v that has chosen a different color. Such games are significant since they model loads of real-worlds scenario with selfish agents and, moreover, they are related to fundamental classes of games. Few results are known concerning the existence of strong equilibria in max k-cut games in this direction. In this paper we make some progress in the understanding of the properties of strong equilibria. In particular, our main result is to show that optimal solutions are 7-strong equilibria. This means that in order a coalition of nodes is able to deviate and drive the system towards a different configuration, i.e. a different coloring, a number of nodes of the coalition strictly larger than 7 is necessary. We also conjecture that, in a generic graph with n nodes, any optimal coloring is also an n-strong equilibrium.