Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.Comment: 26 page

Spatially independent martingales, intersections, and applications

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrised measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the total measure of the intersections is H\"older continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. {\L}aba in connection to the restriction problem for fractal measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in Section 8. Polishing notation and other small changes. All main results unchange

Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. In 1983, Burr and Erdős asked whether the simple lower bound r(Q_n, K_s) ≥ (s−1)(2^(n) − 1) + 1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound

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

The BeamEDM experiment and the measurement of the neutron incoherent scattering length of 199Hg

BeamEDM is a proof-of-principle apparatus to search for a neutron electric dipole moment using a cold neutron beam with a combined Ramsey and time-of-flight technique. Employing a time-of-flight is essential, as it allows to distinguish the v × E systematic effect from an electric dipole moment signal. To date, four beamtimes have been performed both at the Paul Scherrer Institute in Switzerland and at the Institute Laue-Langevin in France. The first part of this thesis presents the development of the apparatus and the measurements performed over the different beamtimes. The nEDM experiment at the Paul Scherrer Institute uses ultra-cold neutrons to measure an electric dipole moment. It requires the use of a mercury co-magnetometer to monitor the magnetic field that the ultra-cold neutrons are probing. Both species can interact with each other via the neutron incoherent scattering length of mercury. This interaction takes the form of a shift in the neutron precession frequency whose sign depends on the mercury atoms' polarization. As the sign of the incoherent scattering length is unknown, the induced shift could be the cause of a systematic effect in the case of a neutron electric dipole measurement. The second part of this thesis details the apparatus, measurement, analysis, and result that has determined the sign of the incoherent scattering length of 199 Hg

Studies of intercellular invasion in vitro using rabbit peritoneal neutrophil granulocytes (PMNS). I. Role of contact inhibition of locomotion.

Intercellular invasion is the active migration of cells on one type into the interiors of tissues composed of cells of dissimilar cell types. Contact paralysis of locomotion is the cessation of forward extension of the pseudopods of a cell as a result of its collision with another cell. One hypothesis to account for intercellular invasion proposes that a necessary condition for a cell type to be invasive to a given host tissue is that it lack contact paralysis of locomotion during collision with cells of that host tissue. The hypothesis has been tested using rabbit peritoneal neutrophil granulocytes (PMNs) as the invasive cell type and chick embryo fibroblasts as the host tissue. In organ culture, PMNs rapidly invade aggregates of fibroblasts. The behavior of the pseudopods of PMNs during collision with fibroblasts was analyzed for contact paralysis by a study of time-lapse films of cells in mixed monolayer culture. In monolayer culture, PMNs show little sign of paralysis of the pseudopods upon collision with fibroblasts and thus conform in their behavior to that predicted by the hypothesis