Partition Properties

Abstract

This chapter describes the progression of ideas and results emerging from partition properties first considered by Erdös and culminating in Silver’s results about the existence of the set of integers 0^#, a principle of transcendence over L. This development incorporated both the refined analysis of combinatorics as well as the full play of model-theoretic techniques, and provided formulations that have come to be regarded as basic. §7 explores partitions of n-tuples, developing the related tree property and further characterizations of weak compactness, and introduces partitions of all finite subsets. §8 gives Rowbottom’s model-theoretic characterizations and results about L, and explores the related concepts of Rowbottom and Jónsson cardinals. Finally, §9 presents Silver’s definitive work on sets of indiscernibles and the implications for L of the existence of 0^#.