Bruce Rothschild

Ramsey’s theorem for

Classes of objects called «-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into r classes of the »c-parameter subsets of any sufficiently large «-parameter set must result in some /-parameter subset with all its /t-parameter subsets in one class. Among the immediate corollaries are the lower dimensional cases of a Ramsey theorem for finite vector spaces (a conjecture of Rota), the theorem of van der Waerden on arithmetic progressions, a set theoretic generalization of a theorem of Schur, and Ramseys Theorem itself.

n

Abstract The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B . For a ϵ A denote by R ( a ) the set { b ϵ B | ( a , b ) ϵ R }. R is called r -Ramsey if for any r -part partition of B there is some a ϵ A with R ( a ) in one part. We investigate questions of whether or not certain R are r -Ramsey where B is a Euclidean space and R is defined geometrically.

-parameter sets

Finding the interacting pairs of proteins between two different protein families whose members are known to interact is an important problem in molecular biology. We developed and tested an algorithm that finds optimal matches between two families of proteins by comparing their distance matrices. A distance matrix provides a measure of the sequence similarity of proteins within a family. Since the protein sets of interest may have dozens of proteins each, the use of an efficient approximate solution is necessary. Therefore the approach we have developed consists of a Metropolis Monte Carlo optimization algorithm which explores the search space of possible matches between two distance matrices. We demonstrate that by using this algorithm we are able to accurately match chemokines and chemokine-receptors as well as the tgfbeta family of ligands and their receptors.

Euclidean Ramsey Theorems. I

The purpose of this paper is to consider a very simple model of the AIDS epidemic. This model illustrates how the spread of AIDS can be affected by parameters whose values, at the present time, are very imprecise or unknown. Of particular interest are the data related to sexual encounters and practices, data which are often unreliable or ambiguous. In this regard, the present model also suggests that the epidemiology of AIDS is particularly sensitive to the limitations in the assessment of sexuality/drug‐related behavior. Using a system of elementary differential equations, the present paper illustrates (using 3 examples) that depending upon the value of critical parameters, one could predict either very rapid increases of seropositivity or a decreasing rate of seropositivi‐ty. Thus, in order to diminish the ambiguity, it becomes crucial to make careful estimates of the sexual “interaction” coefficients, along with all other parameters, to insure that reasonable predictions can be made.

Inferring protein interactions from phylogenetic distance matrices

A family T of k-subsets of an n-set such that no more than r have pairwise fewer than s elements in common is maximum (for sufficiently large n) only if T consists of all the k-sets containing at least one of r fixed disjoint s-subsets.

Sex, drugs and matrices: Mathematical prediction of HIV infection

Equations describing plaque formation in soft agar have been based on certain simplifying assumptions, for which data are presented. The derived equations permit one to calculate (i) average plaque size as a function of the initial density of indicator cells (Do), (ii) the number of cells lysed per plaque as a function of Do, and (iii) the cumulative number of cells lysed at various stages of plaque development. The calculated values agree well with those determined experimentally.

A generalization of the Erdös-Ko-Rado theorem on finite set systems

Abstract In this paper we study generalizations of the following question: Is a subspace of a projective or affine space characterized by the cardinalities of intersections with all hyperplanes? In several cases the answer is affirmative.

Appendix: A model of plaque formation

Introduction. An analogue to a theorem of Ramsey [5] has been conjectured for finite vector spaces by Gian-Carlo Rota. Namely, for each choice of positive integers fe, /, r, and finite field F = GF(q), there exists an integer N(k, /, r\ q) such that if n^N(k, I, r; q) and the ^-dimensional subspaces of an «-dimensional vector space V over F are partitioned into r classes, then some /-dimensional subspace of V has all of its ^-dimensional subspaces in one class. In this note we present a very general theorem of this type, a brief outline of its proof, and general applications, including some cases of Rotas Conjecture. Complete details will appear elsewhere.

Characterizing finite subspaces

This chapter elaborates a restricted version of Hales–Jewetts theorem. A well-known theorem of van der Waerden states that for every pair δ, k of positive integers there exists a positive integer n with the property that for every partition of {1, …, n ) into δ many classes there exists a k -term arithmetic progression contained in one class. In order to obtain a k -term arithmetic progression within one class of the partition a much richer structure (an n -term progression) is partitioned. It is conjectured that for every pair δ, k of positive integers there exists a set A of positive integers that contains no ( k + l )-term progression and still has the property that for every partition of A into δ many classes at least one of the classes contains a k -term progression. It is assumed that A is a finite alphabet, A n is the set of words of length n over A. A n may also be viewed as an n -dimensional cube over A .

Ramsey's theorem for

I worked with Jack van Lint on the JCTA for almost 30 years, beginning in 1976. He was among our most influential and constructive editors. As managing editor I consulted with him regularly on submissions in the areas of designs and codes and practically any other area even remotely connected to his primary interests. Both his interests and his knowledge were extremely broad. Jack’s name appeared as the “Communicating Editor” on many papers, but the number of these does not begin to reflect the amount of work or influence he had. There were countless times we would receive papers and simply consult with Jack. His judgment and wide knowledge were indispensable. He could understand and appreciate papers in very short order, and would be able to help find excellent referees as well as provide sound judgments. It was also characteristic that he would rarely take long to review or comment on papers we sent him. I always felt that Jack was one of the strongest voices in keeping the standards of the Journal high. He had a terrific sense of the significance and quality of the mathematical work we received. He was uncompromising in his values. It was always a great pleasure to be with him. Whenever I had occasion to visit him, he was a gracious host and a wonderful person to work with. At one point, far too early I felt, he told me that he had reached a certain age and that he figured it was time to retire from the Editorial Board. I objected, much to the benefit of the Journal, and he continued to provide his valued advice. That was surely the best “reject” decision the Journal ever made. On behalf of all the editors, I am happy to dedicate this special issue to the memory of Jack van Lint.