Related papers: Joint diamonds and Laver diamonds
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…
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…
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…
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from…
We survey some recent results on the validity of Jensen's diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open…
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 introduce a new parametrized diamond principle denoted $\diamondsuit(\mathsf{LP})$. This principle is akin to the parametrized diamonds of Moore, Hru\v{s}\'ak, and D\v{z}amonja, each of which corresponds to some cardinal invariant of the…
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…
Several authors have introduced various type of coherent-like rings and proved analogous results on these rings. It appears that all these relative coherent rings and all the used techniques can be unified. In [2], several coherent-like…
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,…
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…
We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then…
Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $\kappa$---which we denote $\Diamond_{\text{Lav}}(\kappa)$---asserting the existence of a single function from $\kappa \to H_\kappa$ that behaves much like a…
We prove that for regular $\lambda$ above a strong limit singular $\mu$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $\lambda$ and $\mu$ there are coboundedly many regular…
The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
This paper gives a generic form of the diamond lemma, which includes support for additive and topological structures of the base set, and which does not require any further structure (e.g. an associative multiplication operation) to be…
We prove, e.g., that if lambda=chi^+=2^chi and S subseteq {delta<lambda:cf(delta) neq cf(chi)} is stationary then diamondsuit_lambda holds true.
Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory,…
As noticed in 2006 by the author of the present article, the hypothetical crystal---described by crystallographer F. Laves (1932) for the first time and designated ``Laves' graph of girth ten" by geometer H. S. M. Coxeter (1955)---is a…
We study properties that allow first-order theories to be disjointly combined, including stable infiniteness, shininess, strong politeness, and gentleness. Specifically, we describe a Galois connection between sets of decidable theories,…