Some Independence Results For Amenable Group Actions, Universal Graphs, and Construction Schemes

This thesis uses the technique of forcing to study consistency results in three areas: In the first chapter, we investigate the question of whether or not an amenable subgroup of the permutation group on N can have a unique invariant mean on its action. In joint work with Juris Steprans, we extend the work of Foreman in [13] and show that in the Cohen model such an amenable group with a unique invariant mean must fail to have slow growth rate and a certain weakened solvability condition. In the second chapter, the consistency of a universal graph on 1 with Martins Axiom the negation of the Continuum Hypothesis is investigated. We extend an argument of Shelah in [44] to get a partial result of the consistency of a universal graph on 1 with MA(Cohen), Suslins Hypothesis, and the negation of the Continuum Hypothesis. The third chapter is an investigation of forcing extensions answering some independence questions relating to construction schemes, which are combinatorial schemes for constructing objects with domain introduced by Todorcevic in [50]. In joint work with Fulgencio Lopez, we show that adding 1 Cohen reals adds a capturing construction scheme, that it is consistent to have n-capturing construction schemes but no (n+1)-capturing construction schemes, and show that MA(m-Knaster) and n-capturing are independent if n m and incompatible if n > m.