# Finding $l$ subsets such that their intersection has less or equal than $k$ elements NP-complete or in P?

I have a set $$M$$, subsets $$L_1,…,L_m$$ and natural numbers $$k,l\leq m$$.

The problem is:

Are there l unique indices $$1\leq i_1,…,i_l\leq m$$, such that

$$\hspace{5cm}\left|\bigcap_{j=1}^{l} L_{i_{j}}\right| \leq k$$

Now my question is whether this problem is $$NP$$-complete or not. What irritates me is the two constraints $$l$$ and $$k$$ because the NP-complete problems that were conceptually close to it that I took a look on (set cover, vertex cover) only have one constraint respectively that also appears in this problem.

I then tried to write a polynomial time algorithm that looks at which of the sets $$L_1,…,L_m$$ share more than $$k$$ elements with other sets but even if all sets would share more than $$k$$ elements with other this wouldn’t mean that their intersection has more than $$k$$ elements…

This question kind of comes close but in it there is no restriction on the amount of subsets to use and the size of the intersection should be exactly $$k$$, but maybe this could be useful anyways.

Can somebody further enlighten me ?