Lecture 15 & 16 : Proof of Kadison - Singer Conjecture and the Extensions

Proof sketch. e sketch the main idea of the proof. Let = maxe∈E Reff(e). First we need to construct an isotropic set of vector. For any edge e ∈ E we let ve = L G be. It is easy to see that ∑ e∈E vev T e = I and ‖ve‖ = Reff(e). So, by the above theorem, the edges of G can be partitioned into two sets E1, E2 such that for j ∈ {1, 2}, (1/2−O( √ ))LG LEj (1/2 +O( √ ))LG. The left inequality implies that the effective resistance of each edge of e ∈ E1 with respect to LE1 is about 2Reff(e). So, we can recursively divide E1, E2 into two subgraphs until the effective resistance are close to 1. after log(1/ )− o(log(1/ )) divisions we get to a O( ) thin (connected) subgraph of G.