$\mathrm{◇}$ is equivalent to $\mathrm{♣}$ and continuum hypothesis

If $S$ is a stationary subset of $\kappa$ and implies $2^{\lambda }\le \kappa$ then

$${\mathrm{◇}}_S\leftrightarrow {\mathrm{♣}}_S$$

Moreover, this is best possible: $\mathrm{\neg }{\mathrm{◇}}_S$ is consistent with ${\mathrm{♣}}_S$.

Title	$\mathrm{◇}$ is equivalent to $\mathrm{♣}$ and continuum hypothesis
Canonical name	DiamondIsEquivalentToclubsuitAndContinuumHypothesis
Date of creation	2013-03-22 12:53:55
Last modified on	2013-03-22 12:53:55
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	6
Author	Henry (455)
Entry type	Theorem
Classification	msc 03E45
Synonym	diamond is equivalent to club and continuum hypothesis
Related topic	Diamond
Related topic	Clubsuit