SYMMETRIES IN RAMSEY THEORY

Abstract

In this paper we discuss the role of symmetry in Ramsey theory A recent survey of Ramsey theory by Ne setril appears in to which the reader is referred for a broad view on this subject By contrast this paper will focus almost exclusively on the way in which Ramsey theoretic phenomena are being guided by symmetries Ramsey theory is concerned with the preservation of structure under partitions It therefore deals with constancies or invariances under a very specic kind of change namely the change brought about by a partitioning of certain subobjects of a structure or conguration If one thinks of an observed symmetry as a resistance to change during the ow of a process then it is not surprising that symmetry is a fundamental feature of Ramsey theory Nevertheless the delicate way inwhich Ramsey phenomena are determined by symmetries requires a careful investigation As is well known there is also a highly developed theory dealing with the generation of symmetric patterns and structures under partitions Examples of this theory can be found in the many developments around Van der Waerdens theorem on arithmetic progressions We shall also discuss some results pertaining to this aspect of Ramsey theory in this paper