Split Principles, Large Cardinals, Splitting Families and Split Ideals

We introduce split principles and show that their negations provide simple combinatorial characterizations of large cardinal properties. As examples, we show how inaccessiblity, weak compactness, subtlety, almost ineffability and ineffability can be characterized. Two-cardinal versions of these principles allow us to characterize when the two-cardinal versions of these properties hold: when κ is almost λ-ineffable, λ-ineffable, and we can also characterize mild ineffability by a slightly modified version of the principle. We also characterize when κ is λ-Shelah. Using these facts, we get characterizations of strong compactness and supercompactness. Many of these characterizations can be expressed by saying that certain infinitary splitting numbers are large. The split principles naturally give rise to some large cardinal properties which, in the two-cardinal context, seem to be new. These split principles come with certain ideals, and one of the split principles characterizing a version of κ being λ-Shelah gives rise to a normal ideal on Pκλ. We also explore relationships between these large cardinal concepts and partition relations.