Related papers: Jensen's diamond principle and its relatives
We isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most…
We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its…
We establish the consistency of the failure of the diamond principle on a cardinal $\kappa$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a…
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…
The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously…
I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of…
Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $\kappa$, if $\kappa$ is…
In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset…
The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved under Jensen's Diamond Principle. In this note, the same is verified using the ZFC axioms alone.
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of…
We continue the development of the theory of capturing schemes over $\omega_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum…
Reflection principles (or dually speaking, compactness principles) often give rise to combinatorial guessing principles. Uniformization properties, on the other hand, are examples of anti-guessing principles. We discuss the tension and the…
We introduce and study a family of axioms that closely follows the pattern of parametrized diamonds, studied by Moore, Hru\v{s}\'ak, and D\v{z}amonja in [13]. However, our approach appeals to model theoretic / forcing theoretic notions,…
The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-$\sigma$-scattered. Specifically, we will show that Jensen's principle $\diamondsuit$ implies that there is a minimal…
We deal with the combinatorial principle Weak Diamond, showing that we always either a local version is not saturated or we can increase the number of colours. Then we point out a model theoretic consequence of Weak Diamond.
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
The concepts of closed unbounded (club) and stationary sets are generalised to $\gamma$-club and $\gamma$-stationary sets, which are closely related to stationary reflection. We use these notions to define generalisations of Jensen's…
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond-…
We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the…