Clubsuit

In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski.

Definition
For a given cardinal number $$\kappa$$ and a stationary set $$S \subseteq \kappa$$, $$\clubsuit_{S}$$ is the statement that there is a sequence $$\left\langle A_\delta: \delta \in S\right\rangle$$ such that

$$\clubsuit_{\omega_1}$$ is usually written as just $$\clubsuit$$.
 * every Aδ is a cofinal subset of δ
 * for every unbounded subset $$ A \subseteq \kappa$$, there is a $$\delta$$ so that $$A_{\delta} \subseteq A$$

♣ and ◊
It is clear that ◊ ⇒ ♣, and it was shown in 1975 that ♣ + CH ⇒ ◊; however, Saharon Shelah gave a proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH).