If $\kappa$ is an infinite cardinal then we denote by $[\kappa]^\kappa$ the collection of subsets of $\kappa$ that have cardinality $\kappa$. We say that $\kappa$ is intersectionally strange if there is a cardinal $\lambda \in \kappa$ such that:
there is ${\cal A}\subseteq [\kappa]^\kappa$ with $|{\cal A}| >\kappa$, and $|X\cap Y|<\lambda$ for all $X\neq Y\in {\cal A}$.
Is it consistent that for every infinite cardinal $\kappa$ there is an intersectionally strange cardinal $\alpha$ with $\alpha > \kappa$?
