Semigroup Algebra at Infinity and its Combinatorial Applications

为获得Hindman DMS-0243586奖，首席研究员和他的学生将继续研究离散半群的Stone-Cech紧化的代数结构及其在Ramsey理论中的应用。这两个领域继续取得重大进展，但仍有许多工作要做。在尚未解决的主要代数问题中，是否存在从正整数的Stone-Cech紧化到其余数N*的非平凡连续同态。这个问题可以等价地表述为是否存在N*的一个有限子半群，其成员并不都是幂等元。所提出的代数研究对Ramsey理论具有重要的潜在应用--组合学的一部分，它建立了给定结构的高度正则子结构的存在，这些子结构被划分为有限多个类(或“有限色”)。Stone-Cech紧化的代数在Ramsey理论中最早的应用之一是对以下陈述的简单证明：如果正整数是红色和蓝色的，那么要么存在一个和都是红色的序列(没有重复)，要么存在一个和都是蓝色的序列。更多的应用还在不断被发现。

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.