Equivariant intersection theory is similar to Borel’s equivariant cohomology. The common basic idea is simple. Let X be an algebraic scheme over a field k and let G be an algebraic group acting on X . Since invariant cycles are often too few to get a full-fledged intersection theory (e.g. to have a ring structure in smooth cases) we decide to enlarge this class to include invariant cycles not o...

We show the equivalence of several diierent axiomatizations of the notion of (abstract) probabilistic domain in the category of dcpo's and continuous functions. The axiomatization with the richest set of operations provides probabilistic selection among a nite number of possibilities with arbitrary probabilities, whereas the poorest one has binary choice with equal probabilities as the only ope...

The main result of this paper is that the space of conformally compact Einstein metrics on any given manifold is a smooth, infinite dimensional Banach manifold, provided it is non-empty, generalizing earlier work of Graham-Lee and Biquard. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric a...

Here c+ (6): is a short hand notation (due to Erdiis and Rado) for the following statement: For every partition of the collection of all a-element subsets of a set X of size c, there exists a b-element subset B of C such that all a-element subsets of B belong to one class of the partition. This theorem has been generalized many times and several of these generalizations are both profound and di...

What is a least integer upper bound on van der Waerden number W (r, k) among the powers of the integer r? We show how this can be found by expanding the integer W (r, k) into powers of r. Doing this enables us to find both a least upper bound and a greatest lower bound on W (r, k) that are some powers of r and where the greatest lower bound is equal to or smaller than W (r, k). A finite series ...

Recently [4] we have obtained some new algebraic results about βN, the Stone-Čech compactification of the discrete set of positive integers and about βW , where W is the free semigroup over a nonempty alphabet with infinitely many variables adjoined. (The results about βW extend the Graham-Rothschild Parameter Sets Theorem.) In this paper we derive some Ramsey Theoretic consequences of these re...

The analysis of chessboard pebbling by Andrew Odlyzko is strengthened and generalized, rst to higher dimension and then to arbitrary posets. 1 The pebbling game The pebbling game of Kontsevich is played on the grid points of the rst quadrant. One starts with a single pebble on the origin and a move consists of replacing any pebble with two pebbles, one above and one to the right of the vanishin...

The product of two Schubert cohomology classes on a Grassmannian Grk(C) has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (e.g., the Littlewood-Richardson rule or the more symmetric puzzle rule from A. Knutson, T. Tao, and C. Woodward [KTW]). Recently, W. Graham showed in [G], nonconst...

It was previously shown that any two-colour colouring of K(Cn) must contain a monochromatic planar K4 subgraph for n ≥ N ∗, where 6 ≤ N ∗ ≤ N and N is Graham’s number. The bound was later [2] improved to 11 ≤ N∗ ≤ N . In this article, it is improved to 13 ≤ N∗ ≤ N .

We prove an inequality between three measures of disorder on the symmetric group on n elements. This inequality has been inspired by the well-known Diaconis-Graham inequalities. We also discuss when the inequality is satisfied as equality, and how often this happens. In the case n is odd, the number of permutations that satisfy the equality is a simple function of the Lucas numbers. In addition...