Semigroup Algebra at Infinity and its Combinatorial Applications

摘要为奖励Hindman DMS-0243586首席研究员和他的学生将继续调查的代数结构的斯通-切赫紧化的离散半群及其应用拉姆齐理论。 在这两个领域继续取得重大进展，但仍有许多工作要做。 其中主要的代数问题仍然是未解决的是是否有一个非平凡的连续同态从斯通-切赫紧化的正整数，其其余部分，N*。 这个问题可以等价地表述为是否存在N* 的有限子半群，其成员不都是幂等元。所提出的代数研究对拉姆齐理论有重要的潜在应用--组合学的这一部分建立了给定结构的高度正则子结构的存在性，这些结构被划分为许多类（或“着色”）。 斯通-切赫紧化代数最早应用于拉姆齐理论的一个简单证明是：如果正整数被着色为红色和蓝色，那么要么存在一个序列的所有和（不重复）都是红色，要么存在一个序列的所有和都是蓝色。 更多的应用程序还在继续。

Abstract for award of Hindman DMS-0243586The principal investigator and his students will continue the investigation of the algebraic scructure of the Stone-Cech compactification of a discrete semigroup and its applications to Ramsey Theory. Significant progress continues to be made in both areas, but much remains to be done. Among the major algebraic problems which is still unsolved is whether there is a nontrivial continuous homomorphism from the Stone-Cech compactification of the positive integers to its remainder, N*. That question can be stated equivalently as whether there is a finite subsemgroup of N* whose members are not all idempotents.The proposed algebraic studies have significant potential applications to Ramsey Theory -- that part of combinatorics which establishes the existence of highly regular substructures of given structures that are partitioned into finitely many classes (or "finitely colored"). One of the earliest applications of the algebra of the Stone-Cech compactification to Ramsey Theory was a simple proof of the following statement: If the positive integers are colored red and blue, then there is either a sequence all of whose sums (without repetition) are red or there is a sequence all of whose sums are blue. Additional applications continue to be discovered.