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


1. for every $A\in {\cal C}$ we have $|A|=\kappa$,
2. $|A\_0\cap A\_1|<\lambda$ whenever $A\_0\neq A\_1\in {\cal C}$, and
3. $|{\cal C}| > \kappa$.


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](https://en.wikipedia.org/wiki/Almost_disjoint_sets)). 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 $i(\alpha) < \alpha$?



"Intersection number" of a cardinal
Since this question is still unanswered I thought I might write down some of what you can get out of [Baumgartner's paper](https://pdf.sciencedirectassets.com/272681/1-s2.0-S0003484300X0023X/1-s2.0-0003484376900188/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEEYaCXVzLWVhc3QtMSJIMEYCIQCS2%2FrnWTtDAo53EJ7yg3Rbm3hXn2D%2Fq4J0aht3xYXU2QIhANsXgszYAreGKaEeVdopkgQEkcbkAzMtxjVHNdANdKL1Kr0DCO7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEQAxoMMDU5MDAzNTQ2ODY1IgxGum%2FE8JY0aRPRiBwqkQMGVlby5zE7sWzlY6S92V5hCq5CZvn1%2BvWPykqLb%2FCN1HObGj%2BMCoMJUOYtk0mUI4AuG7e9jHazFgoPWqP3urOwxRRMjJheoezeSn3XVhrcRkQoHDPZLnTX2O5KRDezGafVO8LewdDbZE4rDmRDAoNlWSQLRITJf8WkbQcQjkW9%2BQuDF4VM24xfSNF8ylThGnhy3IDs2Zr48yFmhq8o7c3IF1hm1kAwcnkH%2BNzSqkXM%2BLxaxBSI0lvHkbb%2B%2FF2bIJh4xDQUWm9U83sN1xu9nbqgY0MTww5Nr8tzWtJfO42sW2T8%2BqnEO7HdFWG9C13GC0tVEdsyDasUb7OZvO%2BnyAeXB%2BCxou8yFJPlFuNObnwziDeHF2avvZ04RsVL04e%2B%2BFBSb1yTuNrfBpWR9YrVRfl1CsTTRZyFFjW1e7iS7%2Bl9HMl2Dd3iyigdkP5B6aLFRyV4YeY7rXWfRKdQaAWa97Qk3gLwr8qmSTSX4MqIy%2B6Okto%2BsfSiVHPBfz2S%2By94KJUGMk7NV3jWTaiS82RrV1QTLDC46sv4BTrqAVE%2FGpzWHNB9Tx4byypPU5v7P8zkklIC8wdaH8Jfi2LbPUhoeXLZRVMLRtWXjir0c3%2BXKQdUEuPaiqNtNEX90nFeVR89elYCnaXSBaMzbcJGEwDRWEJ2skf%2FON4WhaOY3hTLjx8zd3vtHJLPx1qryBTIaMcUKtx7Tsu0%2Fz%2BuogTVcDe%2BKq2RxI9epNLPK%2FoXbKfqoLi6RveQucQHMilmn0QltaiVG5LpdKkNd6E1ULXFZ%2Bed6PwjF92OAv3y8E07LRLupiomXxMcOzplMKdwxL2nPSMTjByYReg7NXTdDGafoOnYcU%2ByLr%2BS3w%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20200718T145139Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY4J3XXJVI%2F20200718%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=dc5f0f0445f43a8f0b3417c6ec41288763bb08379b45d33127704e155acf449c&hash=613fe317b0648c04e01c8145c749c0c117c43c579c8f1f4c709e90dae5213863&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=0003484376900188&tid=spdf-410e5fa6-4cbe-42d9-b56d-674d2e94aa8d&sid=990733692415024c016a7f61607907da067fgxrqa&type=client).


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


1. $F\subseteq P(\kappa)$,
2. $|F| = \lambda$,
3. $|X| = \mu$ for all $X\in F$, and
4. $|X\cap Y| < \nu$ 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.


