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.^[1]

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

• every A_δ is a cofinal subset of δ
• for every unbounded subset 👁 {\displaystyle A\subseteq \kappa }
  , there is a 👁 {\displaystyle \delta }
  so that 👁 {\displaystyle A_{\delta }\subseteq A}
  

👁 {\displaystyle \clubsuit _{\omega _{1}}}
is usually written as just 👁 {\displaystyle \clubsuit }
.

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).^[2]

• Club set