UC Irvine

In this dissertation we generalize the supra-SIM property, a partition regular property of subsets of the integers, to a property we call the supra-SBM property of subsets of finitely generated virtually nilpotent groups. We achieve this by defining a measure on the unit ball of asymptotic cones which we show satisfies the Lebesgue density theorem, allowing us to use this measure in place of the Lebesgue measure in the original definition for the integers. These proofs make use of various concepts from nonstandard analysis, measure theory, and combinatorial number theory. We prove that, under an additional assumption, the supra-SBM property is a partition regular property for subsets of finitely generated groups of nilpotency class 2, and conjecture that this holds for all finitely generated virtually nilpotent groups.