Ramsey Theory: Central Sets and Related Combinatorially Rich Sets

这个研究项目涉及拉姆齐理论的数学主题。 拉姆齐理论是组合学的一部分，它处理的问题是，在指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本的最简单的非平凡例子说，只要自然数N的二元子集是双着色的，就一定存在N的某个无限子集，其所有的二元子集都是相同的颜色。拉姆齐理论在数学的各个领域都有应用，也在理论计算机科学、通信和信息理论中有应用。首席研究员将继续指导研究生在有限拉姆齐理论的项目;该奖项仅用于支持研究生，通过参与下一代数学家的研究促进培训。2首席研究员将继续他在拉姆齐理论方面的研究，包括组合学的这一部分与离散半群的Stone-Cech紧化的代数结构之间的相互作用。许多年前，主要研究者证明，只要N是多色的，就一定存在一个颜色的无穷序列及其所有不同项的有限和，而不重复。最初的证明是基本的，但非常复杂。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.格雷泽表明，这一“有限和定理”是一个完全平凡的后果的事实，即斯通-切赫紧化的N可以给出一个代数结构扩展普通除了这是一个紧凑的右拓扑半群，因此有幂等元。具有包含一个序列的所有有限和的性质的集合称为IP集。根据上面发现的联系，一个集合是一个IP集合当且仅当它的闭包中有一个幂等元。那些具有特殊幂等元的集合在其闭包中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集合，并且更容易用初等方式描述。调查这些不同的代数特征的大子集N应继续产生新的拉姆齐理论的结果。

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.