Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

An exploration in Ramsey theory

We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey numbers for graphs requiring more substructure

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

Data and systematic error analysis for the neutron electric dipole moment experiment at the Paul Scherrer Institute and Search for Axionlike Dark Matter

This thesis details work conducted as part of the experimental collaboration responsible for the neutron electric dipole moment (nEDM) experiment based at the Paul Scherrer Institute (PSI). The nEDM is a sensitive probe of a broad range of new CP violating physics beyond the standard model, however it remains elusive: while historic experiments since 1951 have increased in sensitivity by over six orders of magnitude, a nonzero nEDM is yet to be detected. Many theories of physics beyond the standard model predict neutron EDMs of a size that would be detectable by current and next generation experiments, and it has been said that measurements of the neutron EDM have ruled out more theories than any other experiment. One explanation of the smallness of the neutron EDM, the Peccei-Quinn theory, invokes a novel particle, the axion, which is also a credible dark matter candidate. The axion is yet to be detected. The work covers three main areas. First, contributions to the data analysis technique used to analyse the main dataset to produce a new world-leading limit on the neutron EDM. Second, an auxiliary measurement campaign to map the magnetic field within the experiment's magnetic shields is described, and the analysis of these datasets to provide corrections for several critical systematic effects is presented. Finally, a novel analysis of the data taken at a previous-generation nEDM experiment is used to derive the first experimental limits on the coupling of axion-like dark matter particles to gluons is described. These exclusions are up to 1000 times stronger than previous results for cosmologically interesting 10-22 eV axions

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b

Efficient Subsidization of Human Capital Accumulation with Overlapping Generations and Endogenous Growth

This paper studies second best policies for education, saving, and labour in an OLG model in which endogenous growth results from human capital accumulation. Government expenditures have to be financed by linear instruments so that growth equilibria are inefficient. The inefficiency is exacerbated if selfish individuals externalize the positive effect of education on descendents’ productivity. It is shown to be second best to subsidize education even relative to the first best if the elasticity of the human capital investment function is strictly increasing.OLG model; endogenous growth; endogenous labour, education, and saving; intergenerational externalities; optimal taxation

Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress