The following notion is introduced by Assaf Rinot:


**Definition.** A singular cardinal $\kappa$ is a prevalent singular
cardinal iff there exists a family $\mathbb{A}\subset P(\kappa)$ with $|\mathbb{A}| = \kappa$
and $sup\{|A| : A\in \mathbb{A}\} < \kappa$ such that any $B\subset \kappa$ with
$|B| < cf(\kappa)$ is contained in some $A\in \mathbb{A}$.


**Prevalent Singular Cardinals Hypothesis (PSCH)** states that any singular cardinal is a prevalent singular cardinal.


Clearly any singular cardinal of countable cofinality is prevalent.


**Question.** What is known about the consistency of the failure of **PSCH**?


**Remark 1.** **PSCH** follows from $GCH, SCH, PFA$ and many other combinatorial principles. 


**Remark 2.** As stated in Assaf Rinot's paper "On topological spaces of singular density and minimal weight, Topology and its Applications 155 (2007) 135–140", $PSCH$ is a very weak assertion and all currently known methods for violating statements
of similar flavor, will fail to violate the $PSCH$.


**Question 2.** Is there any relation between **PSCH** and some combinatorial principles introduced by Shelah in PCF theory like $Cov$ or ...?



Prevalent singular cardinals hypothesis
A singular cardinal $\lambda$ is prevalent iff there exists some cardinal $\mu<\lambda$ such that $Cov(\lambda,\mu,cf(\lambda),2)=\lambda$. In his solution of the pcf conjecture, Gitik has constructed a model where there exists a singular cardinal $\lambda$ of cofinality $\aleph\_1$ such that $pp(\mu)>\lambda$ for cofinally many $\mu<\lambda$ of countable cofinality. Given $\mu<\lambda$, pass to a bigger $\mu'<\lambda$ of countable cofinality with $pp(\mu')>\lambda$. Then, we have $pp(\mu')\le Cov(\mu',\mu',cf(\mu')^+,2)\le Cov(\lambda,\mu',cf(\lambda),2)\le Cov(\lambda,\mu,cf(\lambda),2)$. So $Cov(\lambda,\mu,cf(\lambda),2)>\lambda$ for every $\mu<\lambda$, meaning that $\lambda$ is not prevalent.


