Was Ulam right? I: Basic theory and subnormal ideals

Joint work with Tanmay Inamdar.

Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow to characterize large cardinals, most notably weakly compact and ineffable cardinals. We also develop the basic theory of these coloring principles, connecting them to the classical negative square bracket partition relations, proving pumping-up theorems, and deciding various instances of theirs.

We also demonstrate that our principles provide a uniform way of obtaining non-saturation results ideals satisfying a property we call subnormality in contexts when Ulam matrices might not be available.

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