1 4 A ug 2 00 6 Club guessing and the universal models Mirna Džamonja School of Mathematics University of East Anglia Norwich , NR 4 7 TJ , UK

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element. 1 0 Introduction A natural problem in mathematics is the following: given some partially ordered or a quasi-ordered set or a class, is there the largest element in it. An aspect of this question appears in the theory of order where one concentrates on the properties of the set and the partial order, ignoring the properties of the individual elements of the set. A very different view of this question is obtained when one takes the point that it is the structure of individual elements that is of interest. An instance of this is the question of universality. Here we are given a class or a set of objects and a notion of embedding between them, and we ask if there is an object in the class that 1 The author thanks EPSRC for their support on as Advanced Fellowship. AMS 2000 Classification: 03C55, 03E04, 03C45.