"Intersection number" of a cardinal Let $\kappa$ be an infinite cardinal. We call a cardinal $\lambda \leq 2^\kappa$ intersecting if there is ${\cal C}\subseteq {\cal P}(\kappa)$ such that

*

*for every $A\in {\cal C}$ we have $|A|=\kappa$,

*$|A_0\cap A_1|<\lambda$ whenever $A_0\neq A_1\in {\cal C}$, and

*$|{\cal C}| > \kappa$.

We denote the smallest intersecting cardinal of $\kappa$ by $i(\kappa)$. For instance we have $i(\aleph_0) = \aleph_0$ (also see the concept of an almost disjoint family). By the comments of users bof and Alessandro Codenotti, we always have $i(\kappa) \leq \kappa$ for any infinite cardinal $\kappa$.
Question. If $\kappa$ is an infinite cardinal, is there a cardinal $\alpha\geq\kappa$ with $i(\alpha) < \alpha$?
 A: Since this question is still unanswered I thought I might write down some of what you can get out of Baumgartner's paper.
In Baumgartner's notation (see the beginning of section 2), $A(\kappa,\lambda,\mu,\nu)$ means that there exists a family of sets $F$ such that

*

*$F\subseteq P(\kappa)$,

*$|F| = \lambda$,

*$|X| = \mu$ for all $X\in F$, and

*$|X\cap Y| < \nu$ for all $X,Y\in F$ with $X\neq Y$.

Hence the connection is that $\lambda$ is intersecting (in your notation) if and only if $A(\kappa,\kappa^+,\kappa,\lambda)$ holds.
In Theorem 3.4(a) Baumgartner proves that, assuming GCH, for any cardinals $\nu \le \mu \le \kappa$, $A(\kappa,\kappa^+,\mu,\nu)$ holds if and only if $\mu = \nu$ and $cf(\mu) = cf(\kappa)$.  Since we're only interested in the case where $\mu = \kappa$, this implies that, under GCH, $i(\kappa) = \kappa$ for all $\kappa$.  Note that this conclusion already follows from bof's comments.
The other side is partly covered by Theorem 6.1, which says: assuming GCH holds in $V$, for any cardinals $\nu \le \kappa \le \lambda$ such that $\nu$ is regular, there is a forcing extension $V[G]$ which preserves the cofinalities (hence cardinals) of $V$, in which $A(\kappa,\lambda,\kappa,\nu)$ is true.  Hence you can make $i(\kappa) = \omega$ true for any particular $\kappa$, starting from a model of GCH.
It remains to show the consistency of the statement in your question, i.e. for all $\kappa$ there is some $\alpha \ge \kappa$ such that $i(\alpha) < \alpha$.  Maybe someone who knows about class forcing can step in.
