Structural reflection for large cardinal partition properties

Abstract

In the theory of large cardinals, the Structural Reflection research program has the ultimate goal of providing a uniform way of characterizing any large cardinal notion in terms of structural reflection principles. In the present work, we study and provide such a characterization for Erdős, Ramsey, Rowbottom and Jónsson cardinals, which are large cardinal notions commonly defined in terms of partition properties and contained in the region below the first measurable cardinal. We introduce three new families of structural reflection principles: the invariant structural reflection principles, which characterize Erdős and Ramsey cardinals; the two-cardinal structural reflection principles, which characterize Rowbottom cardinals; and the proper structural reflection principles, which characterize Jónsson cardinals. Finally, we show how a particular generalization of a proper structural reflection principle yields a characterization of exacting cardinals.