Ramsey Theory: Central sets and related combinatorially rich sets

拉姆齐理论是组合学的一部分，它处理的问题是，在一个特定集合的有限分割的某个单元中（或者有时在任何适当的“大”子集中），人们可以期望找到什么样的齐次结构。例如，拉姆齐定理的无限版的最简单的非平凡实例说，每当正整数集合N的两个元素子集有有限颜色时，必然存在N的某个无限子集，它的两个元素子集都是相同的颜色。许多年前，首席研究员证明了，当N是有限色时，在一种颜色中必须存在一个无限序列及其所有不同项的有限和而不重复。最初的证明很简单，但很复杂。随后，人们发现了其他不那么复杂的证明。但是在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论，证明N的Stone-Cech紧化可以给出一个扩展普通加法的代数结构，使其成为紧的右拓扑半群，因此具有幂等。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。根据上面发现的联系，一个集合是IP集合当且仅当它在n的stone - ech紧化的闭包中具有幂等幂等。那些在闭包中具有称为“极小”的特殊幂等幂等的集合是“中心”集合。这些集合具有更强的性质，其中许多是中心集合定理的结果。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”，它们更容易用初等的方式来描述。对这些不同代数特征的N的大子集的研究应该会继续产生新的拉姆齐理论结果。这笔资金的很大一部分将用于支持霍华德大学的研究生，这是一所历史悠久的黑人大学。特别值得一提的是，该基金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中代表性严重不足的人口的数学家。

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.