Ramsey Theory: Central sets and related combinatorially rich sets

拉姆齐理论是组合学的一部分，它处理人们可以期望在特定集合的有限划分的某个单元中(或者有时在任何适当的“大”子集中)找到什么样的同质结构的问题。例如，拉姆齐定理无限版本的最简单的非平凡实例是这样的：只要正整数集N的两个元素子集是有限着色的，就一定存在N的某个无限子集，其所有两个元素子集都是相同的颜色。许多年前，首席研究者证明了当N是有限着色时，在一种颜色中一定存在一个无穷序列以及它的所有不同项的有限和而没有重复。最初的证明是基本的，但非常复杂。随后，又发现了其他不那么复杂的证据。但在1975年，F.Galvin和S.Glazer证明了这个“有限和定理”是一个完全平凡的结果，因为N的Stone-Cech紧化可以被赋予一个扩展普通加法的代数结构，使它成为紧的右拓扑半群，从而具有幂等元。具有包含一个序列的所有有限和的性质的集合称为IP集合。借助于上面发现的联系，一个集合是一个IP集合当且仅当它在N的Stone-Cech紧化中的闭包中有一个幂等元。那些在其闭包中有特殊的幂等元被称为“极小”的集合是“中心”集。这些集合具有更强的性质，其中许多性质是中心集定理的结果。但是中心集有一个非常复杂的基本描述。满足中心集定理结论的集合称为“C-集合”，更容易用初等的方式来描述。对N的这些不同的代数刻画的大子集的拟议研究应该会继续产生新的Ramsey理论结果。这笔赠款中的很大一部分资金将为霍华德大学的研究生提供支持，霍华德大学是一所历史悠久的黑人大学。特别是，这笔拨款将为三名博士生提供津贴，其中两名是美国黑人，均为女性。因此，该项目将有助于培养来自美国数学家人口中比例严重偏低的数学家。

Ramsey Theory is that part of combinatorics that deals with the question of what sort of homogeneous structures one can expect to find in some one cell of a finite partition of a specified set (or sometimes in any suitably "large" subset). For example, the simplest nontrivial instance of the infinite version of Ramsey's Theorem says that whenever the two-element subsets of the set N of positive integers are finitely colored, there must be some infinite subset of N all of whose two element subsets are the same color. Many years ago, the principal investigator proved that whenever N is finitely colored, there must exist in one color an infinite sequence together with all of its finite sums of distinct terms without repetition. The original proof was elementary, but very complicated. Subsequently, other proofs were found that were less complicated. But in 1975, F. Galvin and S. Glazer showed that this "Finite Sums Theorem" is a completely trivial consequence of the fact that the Stone-Cech compactification of N can be given an algebraic structure extending ordinary addition which makes it a compact right topological semigroup, and therefore has idempotents. Sets with the property that they contain all the finite sums from a sequence are called IP sets. By virtue of the connection discovered above, a set is an IP set if and only if it has an idempotent in its closure in the Stone-Cech compactification of N. Those that have special idempotents which are called "minimal" in their closure are "central" sets. These sets have much stronger properties, many of which are consequences of the Central Sets Theorem. But central sets have a very complicated elementary description. Sets which satisfy the conclusion of the Central Sets Theorem are called "C-sets", and are much easier to describe in an elementary fashion. The proposed investigation of these various algebraically characterized large subsets of N should continue to yield new Ramsey-theoretic results. A significant portion of the funds in this grant will provide support for graduate students at Howard University, an historically black university. In particular, the grant will provide stipends for three Ph.D. students, two of whom are black Americans, both female. This project will therefore be instrumental in training mathematicians who come from a population that is severely underrepresented within the population of US mathematicians.