Was Ulam right? I: Basic theory and subnormal ideals

We introduce various colouring principles which generalise the so-called onto mapping principle of Sierpiński to larger cardinals and general ideals. prove that these capture notion an Ulam matrix allow characterise large cardinals, most notably weakly compact ineffable cardinals. also develop basic theory principles, connecting them classical negative square bracket partition relations, proving pumping-up theorems, deciding instances theirs. demonstrate our provide a uniform way obtaining non-saturation results for ideals satisfying property we call subnormality in contexts where matrices might not be available.