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.
