Upper density problems in infinite Ramsey theory

We consider the following question in infinite Ramsey theory, introduced by Erdős and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H. The problem of determining the Ramsey density of the infinite path was first studied by Erdős and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2. We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von Erdős und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzählbar unendlicher Graph. Welches ist der größtmögliche Wert λ, sodass wir, wenn die Kanten des vollständigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefärbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H. Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von Erdős und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhängigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte für eine Vielzahl von Graphen, darunter alle lokal endlichen Wälder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren für ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2. Darüber hinaus untersuchen wir eine Variante des oben beschriebenen Problems für Listenfärbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollständigen Graphen mit Knotenmenge N eine Liste der Größe Zwei zuzuweisen, sodass in jeder zugehörigen Listenfärbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren

Satisfaction degree of elderly fully edentulous patients after insertion of a new fixed or removable full denture An integrative systematic review

Introdução: Progressivamente ao longo dos anos, o número de idosos tem vindo a aumentar. Nesta faixa etária, os problemas dentários tendem a ser mais persistentes e severos, por terem sido muitas das vezes negligenciamos em prol de outras urgências sanitárias. Neste sentido, é crucial compreender como melhorar a atual qualidade de vida e determinar se os problemas e tratamentos dentários têm um impacto sobre ela. Objectivo: Avaliar o grau de satisfação de pacientes idosos totalmente edêntulos após a inserção de uma nova prótese total fixa ou removível. Materiais e métodos: Foi realizada uma pesquisa bibliográfica nas bases de dados Pubmed, Scielo, Cochrane, Science direct e EBSCO, utilizando a seguinte combinação de termos de pesquisa "Quality of life" OR "Satisfaction" AND "Elderly" OR "Aged"AND "Complete denture" AND "Edentulous". Resultados: Foram seleccionados um total de 17 artigos publicados entre os anos 2012 e 2023, por cumprirem os critérios previamente definidos. Discussão: Perceber os factores psicossociais que afectam a eficácia dos cuidados terapêuticos e a qualidade de vida dos pacientes edêntulos é crucial para melhorar o seu bem-estar. Estudos demonstraram que a satisfação do paciente com as novas dentaduras completas está significativamente associada a uma melhor qualidade de vida relacionada com a saúde oral, enquanto as próteses sobre implantes podem proporcionar maior estabilidade, força de mordida e satisfação, levando a uma melhor qualidade de vida global. Conclusão: A qualidade de vida da população idosa total edêntula poderia ser melhorada através da realização de novas próteses totais.Introduction: Progressively over the years, the number of elderly people has been increasing. In this age group, dental issues tend to be more persistent and severe, because they have often been neglected in favour of other health emergencies. In this sense, it is crucial to understand how to improve the current elderly people quality of life and determine whether dental issues and treatments have an impact on it. Aim: Evaluate the satisfaction degree of elderly fully edentulous patients after insertion of a new fixed or removable full denture. Materials and methods: A literature search was conducted in the Pubmed, Scielo, Cochrane, Science direct and EBSCO databases, using the following combination of search terms "Quality of life" OR "Satisfaction" AND "Elderly" OR "Aged "AND "Complete denture" AND "Edentulous". Results: Were selected a total of 17 articles, published between the years 2012 and 2023, in order to meet the previously defined criteria. Discussion: Understanding the psychosocial factors that impact the effectiveness of therapeutic care and quality of life for edentulous patients is crucial for improving their well-being. Studies have shown that patient satisfaction with new complete dentures is significantly associated with improved oral health-related quality of life, while implant overdentures can provide enhanced stability, bite force and satisfaction, leading to improved overall quality of life. Conclusion: The quality of life of the total edentulous elderly population could be improved through the realisation of new total prostheses

A Note on the Minimum Number of Edges in Hypergraphs with Property O

An oriented $k$-uniform hypergraph is said to have Property O if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. Recently Duffus, Kay and R\"{o}dl investigated the minimum number $f(k)$ of edges in a $k$-uniform hypergaph with Property O. They proved that $k! \leq f(k) \leq (k^2 \ln k) k!$, where the upper bound holds for $k$ sufficiently large. In this short note we improve their upper bound by a factor of $k \ln k$, showing that $f(k) \le \left(\lfloor \frac{k}{2} \rfloor +1 \right) k! - \lfloor \frac{k}{2} \rfloor (k-1)!$ for every $k\geq 3$. We also show that their lower bound is not tight. Furthermore, Duffus, Kay and R\"{o}dl also studied the minimum number $n(k)$ of vertices in a $k$-uniform hypergaph with Property O. For $k=3$ they showed $n(3) \in \{6,7,8,9\}$, and asked for the precise value of $n(3)$. Here we show $n(3)=6$.Comment: 6 pages, 1 figur

Planar graph with twin-width seven

We construct a planar graph with twin-width equal to seven

Lobbying for agroforestry in the EU

Presentatio

Inferring useful static types for duck typed languages

Complete and precise identification of types is essential to the effectiveness of programming aids such as refactoring or code completion. Existing approaches that target dynamically typed languages infer types using flow analysis, but flow analysis does not cope well with heavily used features such as heterogeneous containers and implicit interfaces. Our solution makes the assumption that programs that are known to work do not encounter run-time type errors which allows us to derive extra type information from the way values are used, rather than simply where those values originate. This is in keeping with the “duck typing” philosophy of many dynamically typed languages. The information we derive must be conservative, so we describe and formalise a technique to ‘freeze’ the duck type of a variable using the features, such as named methods, that are provably present on any run of the program. Development environments can use these sets of features to provide code-completion suggestions and API documentation, amongst other things. We show that these sets of features can be used to refine imprecise flow analysis results by using the frozen duck type to perform a structural type-cast. We first formalise this for an idealised duck-typed language semantics and then show to what extent the technique would work for a real-world language, Python. We demonstrate its effectiveness by performing an analysis of several real-world Python programs which shows that we can infer the types of method-call receivers more precisely than can flow analysis alone

The dimension of the region of feasible tournament profiles

Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of the region of $k$-vertex graph profiles, i.e., the region of feasible densities of $k$-vertex graphs in large graphs, is equal to the number of non-trivial connected graphs with at most $k$ vertices. We determine the dimension of the region of $k$-vertex tournament profiles. Our result, which explores an interesting connection to Lyndon words, yields that the dimension is much larger than just the number of strongly connected tournaments, which would be the answer expected as the analogy to the setting of graphs