Two extensions of Ramsey's theorem

Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of R\"odl, we show that there is a constant $c>0$ such that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\} contains a monochromatic clique S for which the sum of 1/\log i over all vertices i \in S is at least c\log\log\log n. This is tight up to the constant factor c and answers a question of Erd\H{o}s from 1981. Motivated by a problem in model theory, V\"a\"an\"anen asked whether for every k there is an n such that the following holds. For every permutation \pi of 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2, ..., n} contains a monochromatic clique a_1<...<a_k with a_{\pi(1)+1}-a_{\pi(1)}>a_{\pi(2)+1}-a_{\pi(2)}>...>a_{\pi(k-1)+1}-a_{\pi(k-1)}. That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erd\H{o}s, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.Comment: 21 pages, accepted for publication in Duke Math.

Commodity Taxation and Social Welfare: The Generalised Ramsey Rule

Commodity taxes have three distinct roles: (1) revenue collection, (2) interpersonal redistribution, and (3) resource allocation. The paper presents an integrated treatment of these three concerns in a second-best general equilibrium framework, which leads to the "generalised Ramsey rule for optimum taxation. We show how many standard results on optimum taxation and tax reform have straightforward counterpart in this general framework. Using this framework, we also try to clarify the notion of "deadweight loss", as well as the relation between alternative distributional assumptions and the structure of optimum taxes.

An Improved Neutron Electric Dipole Moment Experiment

A new measurement of the neutron EDM, using Ramsey's method of separated oscillatory fields, is in preparation at the new high intensity source of ultra-cold neutrons (UCN) at the Paul Scherrer Institute, Villigen, Switzerland (PSI). The existence of a non-zero nEDM would violate both parity and time reversal symmetry and, given the CPT theorem, might lead to a discovery of new CP violating mechanisms. Already the current upper limit for the nEDM (|d_n|<2.9E-26 e.cm) constrains some extensions of the Standard Model. The new experiment aims at a two orders of magnitude reduction of the experimental uncertainty, to be achieved mainly by (1) the higher UCN flux provided by the new PSI source, (2) better magnetic field control with improved magnetometry and (3) a double chamber configuration with opposite electric field directions. The first stage of the experiment will use an upgrade of the RAL/Sussex/ILL group's apparatus (which has produced the current best result) moved from Institut Laue-Langevin to PSI. The final accuracy will be achieved in a further step with a new spectrometer, presently in the design phase.Comment: Flavor Physics & CP Violation Conference, Taipei, 200

Commodity Taxation and Social Welfare: The Generalised Ramsey Rule

Commodity taxes have three distinct roles: (1) revenue collection, (2) interpersonal redistribution, and (3) resource allocation. The paper presents an integrated treatment of these three concerns in a second-best general equilibrium framework, which leads to the 'generalised Ramsey rule' for optimum taxation. We show how many standard results on optimum taxation and tax reform have a straightforward counterpart in this general framework. Using this framework, we also try to clarify the notion of 'deadweight loss' as well as the relation between alternative distributional assumptions and the structure of optimum taxes.Commodity taxation, efficiency, redistribution, shadow prices

Ramsey numbers involving a triangle: theory and algorithms

Ramsey theory studies the existence of highly regular patterns in large sets of objects. Given two graphs G and H, the Ramsey number R(G, H) is defined to be the smallest integer n such that any graph F with n or more vertices must contain G, or F must contain H. Albeit beautiful, the problem of determining Ramsey numbers is considered to be very difficult. We focus our attention on efficient algorithms for determining Ram sey numbers involving a triangle: R(K3 , G). With the help of theoretical tools, the search space is reduced by using different pruning techniques and linear programming. Efficient operations are also carried out to mathematically glue together small graphs to construct larger critical graphs. Using the algorithms developed in this thesis, we compute all the Ramsey numbers R(Kz,G), where G is any connected graph of order seven. Most of the corresponding critical graphs are also constructed. We believe that the algorithms developed here will have wider applications to other Ramsey-type problems

Partition Theorems for Spaces of Variable Words

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135422/1/plms0449.pd