Was Ulam right? I: Basic theory and subnormal ideals

TL;DR

This paper introduces new colouring principles that generalise classical set-theoretic concepts, characterise large cardinals, and provide tools for analyzing ideals and partition relations in set theory.

Contribution

It develops a unified framework of colouring principles that generalise Ulam matrices, characterise large cardinals, and connect to classical partition relations and ideal saturation.

Findings

Characterisation of weakly compact and ineffable cardinals via colouring principles.

Development of pumping-up theorems for these principles.

Application to non-saturation results for subnormal ideals.

Abstract

We introduce various colouring principles which generalise 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 characterise large cardinals, most notably weakly compact and ineffable cardinals. We also develop the basic theory of these colouring 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 for ideals satisfying a property we call subnormality in contexts where Ulam matrices might not be available.