A measurable selection theorem

Measurable Utility and the Measurable Choice Theorem

The Measurable Kesten Theorem

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigi...

Brooks’s Theorem for Measurable Colorings

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, suc...

The Halpern-läUchli Theorem at a Measurable cardinal

Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern-Läuchli Theorem on d many normal κ-trees at a measurable cardinal κ, giv...

A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations