Winter School in Abstract Analysis, January 2023

I gave a 3-lecture tutorial at the Winter School in Abstract Analysis in Steken, January 2023.

Title: Club guessing

Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colorings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more.
In this tutorial, we will give a gentle introduction to the subject.

In Part I, we shall present the basic club-guessing principle $CG(S)$, and prove that it holds for $S:=E_{\theta }^{\kappa }$ (the set of all ordinals below $\kappa$ of cofinality $\theta$) provided that $\theta <\kappa$ are regular uncountable cardinals and ${\theta }^{+}<\kappa$. An ad-hoc treatment of the case $\theta ={\mathrm{\aleph }}_0$ will be given, and raising the concept of relative club-guessing. We shall also present postprocessing functions, and compare Ulam matrices with a coloring obtained by Sirepinski under a GCH-type assumption, motivating an application of club-guessing that will appear in the second lecture.

In Part II, we shall construct an optimal Sirepinski-type coloring in ZFC, using pcf scales and club-guessing. We then move on to address the extreme case of $CG(E_{\theta }^{\kappa })$ where $\kappa ={\theta }^{+}$. This inevitably leads to the introduction of the idealized guessing principle $CG(S,\overrightarrow{J})$, which in turn requires the study of amenable $C$-sequences and nonconservative postprocessing functions. As an application, we waive an uncountability assumption from the first lecture, proving that $CG(E_{\theta }^{\kappa })$ holds for every pair $\theta <\kappa$ of infinite regular cardinals such that ${\theta }^{+}<\kappa$.

In Part III, we combine idealized club-guessing with relative guessing, resulting in the principle $CG(S,T,\overrightarrow{J})$. We explain how partitioned club-guessing can compensate for the inevitable shift to idealized $CG$. We show how anti-large-cardinal hypotheses on $\theta$ or on $\kappa$ imply that a given witness to $CG(E_{\theta }^{\kappa },T,\overrightarrow{J})$ may be partitioned, and examine the impact of the sequence of ideals $\overrightarrow{J}$ on one’s ability to partition. This leads to a Solovay-type decomposition theorem for club-guessing and to a general discussion on how to move from $CG(S,T,\overrightarrow{J})$ to $CG(S^{\prime },T^{\prime },\overrightarrow{J^{\prime }})$. Combining all the acquired machinery, we conclude that for every cardinal $\lambda \ge {\beth }_{\omega }$ such that $\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}({\lambda }^{+})$ holds, $CG(S,T,⟨J^{\text{bd}}[\delta ]\mid \delta \in S⟩)$ holds for all stationary subsets $S,T$ of ${\lambda }^{+}$.

Lecture 1 ** Lecture 2 ** Lecture 3