Ramsey Theory: Central Sets and Related Combinatorially Rich Sets

本研究项目涉及拉姆齐理论的数学课题。拉姆齐理论是组合学的一部分，它处理的问题是，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），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，当自然数N的两个元素子集是有限色时，必然存在N的某个无限子集，其所有的两个元素子集都是相同的颜色。拉姆齐理论在数学的各个领域以及理论计算机科学、通信和信息理论中都有广泛的应用。首席研究员将继续指导研究生进行有限拉姆齐理论的项目；该奖项仅包括对研究生的支持，通过参与研究促进下一代数学家的培训。首席研究员将继续他在Ramsey理论方面的研究，包括这部分组合学与离散半群的stone - ech紧化代数结构之间的相互作用。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，证明了N的Stone-Cech紧化可以给定一个扩展普通加法的代数结构，它是紧的右拓扑半群，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。根据上面发现的连接，当且仅当一个集合的闭包是幂等的，它就是一个IP集合。那些具有特殊的幂等函数，在闭包中称为极小值的集合是中心集。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。满足中心集合定理结论的集合被称为C集合，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。

This research project concerns the mathematical subject of Ramsey theory. 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 specified sets (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 natural numbers N are finitely colored, there must be some infinite subset of N all of whose two-element subsets are the same color. Ramsey theory has applications in a wide variety of areas of mathematics as well as in theoretical computer science, communications, and information theory. The principal investigator will continue to mentor graduate students in projects on finite Ramsey Theory; this award consists solely of support for graduate students, facilitating the training through research involvement of the next generation of mathematicians.The principal investigator will continue his research in Ramsey theory, including the interactions between this part of combinatorics and the algebraic structure of the Stone-Cech compactification of a discrete semigroup. The principal investigator proved many years ago 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 is 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. 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 that satisfy the conclusion of the Central Sets Theorem are called C sets, and are much easier to describe in an elementary fashion. Investigation of these various algebraically characterized large subsets of N should continue to yield new Ramsey theoretic results.