$\mathrm{♣}$

${\mathrm{♣}}_S$ is a combinatoric principle weaker than ${\mathrm{◇}}_S$. It states that, for $S$ stationary in $\kappa$, there is a sequence ${⟨A_{\alpha }⟩}_{\alpha \in S}$ such that $A_{\alpha }\subseteq \alpha$ and $\sup (A_{\alpha })=\alpha$ and with the property that for each unbounded subset $T\subseteq \kappa$ there is some $A_{\alpha }\subseteq X$.

Any sequence satisfying ${\mathrm{◇}}_S$ can be adjusted so that $\sup (A_{\alpha })=\alpha$, so this is indeed a weakened form of ${\mathrm{◇}}_S$.

Any such sequence actually contains a stationary set of $\alpha$ such that $A_{\alpha }\subseteq T$ for each $T$: given any club $C$ and any unbounded $T$, construct a $\kappa$ sequence, $C^{*}$ and $T^{*}$, from the elements of each, such that the $\alpha$-th member of $C^{*}$ is greater than the $\alpha$-th member of $T^{*}$, which is in turn greater than any earlier member of $C^{*}$. Since both sets are unbounded, this construction is possible, and $T^{*}$ is a subset of $T$ still unbounded in $\kappa$. So there is some $\alpha$ such that $A_{\alpha }\subseteq T^{*}$, and since $\sup (A_{\alpha })=\alpha$, $\alpha$ is also the limit of a subsequence of $C^{*}$ and therefore an element of $C$.

Title	$\mathrm{♣}$
Canonical name	clubsuit
Date of creation	2013-03-22 12:53:52
Last modified on	2013-03-22 12:53:52
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	4
Author	Henry (455)
Entry type	Definition
Classification	msc 03E65
Synonym	clubsuit
Related topic	Diamond
Related topic	DiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic	ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic	CombinatorialPrinciple