Jacob Davis successfully defended his dissertation, “Universal Graphs at ℵ𝜔1+1 and Set-theoretic Geology,” at Carnegie Mellon University on April 29, 2016, under the supervision of James Cummings. I was on the dissertation committee (participating via Google Hangouts), along with Ernest Schimmerling and Clinton Conley.

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The thesis consisted of two main parts. In the first half, starting from a model of ZFC with a supercompact cardinal, Jacob constructed a model in which 2ℵ𝜔1 =2ℵ𝜔1+1 =ℵ𝜔1+3 and in which there is a jointly universal family of size ℵ𝜔1+2 of graphs on ℵ𝜔1+1.  The same technique works with any uncountable cardinal in place of 𝜔1.  In the second half, Jacob proved a variety of results in the area of set-theoretic geology, including several instances of the downward directed grounds hypothesis, including an analysis of the chain condition of the resulting ground models.