Sauer Problems for Set Systems

9401351 Radcliffe该奖项用于资助Andrew Radcliffe教授对绍尔定理及其含义的研究。雷德克里夫教授打算研究反链的踪迹、一般踪迹问题和反链。这项工作应该对学习理论、经验过程和巴拿赫空间理论有启示。这项研究属于组合学的广泛范畴，这是当今数学中最活跃的领域之一。从根本上说，组合学是对系统计数的研究，尽管这一研究更多地涉及对计数结果的比较。绍尔定理是一个一般的不等式，它允许人们知道一个集合比另一个集合包含更多的对象。当然，集合必须是特定类型的，但这些条件并不难满足。虽然它的根源可以追溯到几个世纪以前，但组合学领域在过去的几十年里有了爆炸性的发展。这种增长来自于它在通信和信息技术中的重要性，以及在这项工作中，人工智能。

9401351 Radcliffe This award funds the research of Professor Andrew Radcliffe into Sauer's theorem and its implications. Prof. Radcliffe intends studying traces of antichains, general trace problems and antichains. The work should have implications for learning theory, empirical processes and the theory of Banach spaces. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. At its roots, combinatorics is the study of systematic counting, although this research is more involved with comparing the results of counting. Sauer's theorem is a general inequality that allows one to know that one collection contains more objects than another. Of course the collections must be of particular types, but the conditions are not hard to satisfy. Although its roots go back several centuries, the field of combinatorics has had an explosive development in the past few decades. This growth comes from its importance in communications and information technology, and as in this work, artificial intelligence.