Discrepancy, Graphs, and the Kadison–Singer Problem

where R ⊂ [n] denotes the set of red elements. In other words, it is possible to partition [n] into two subsets so that this partition is very close to balanced on each one of the test sets Si. Note that a “continuous” partition which splits each element exactly in half will be exactly balanced on each Si; the content of Spencer’s theorem is that we can get very close to this ideal situation with an actual, discrete partition which respects the wholeness of each element. Spencer’s theorem and its variants have had applications in approximation algorithms, numerical integration, and many other areas. In this post I will describe a new discrepancy theorem [1] due to Adam Marcus, Dan Spielman, and myself, which also seems to have many applications. The theorem is about “uniformly” partitioning sets of vectors in R and says the following: