Set systems with "small" intersections

Question

Are there infinite cardinals $\kappa < \lambda$ such that here is a collection ${\cal A}$ of subsets of $\lambda$ with the following properties:

1. $|{\cal A}| = 2^\lambda$, and

2. $A\neq B\in {\cal A}$ implies $|A\cap B|\leq \kappa$.

Answer 1

Here's a quick observation: suppose $2^\kappa=\kappa^+$. Then we can mimic the construction of an almost disjoint set of sets of natural numbers of size continuum: to each map $p: \kappa^+\rightarrow 2$, we associate the set of small approximations $A_p=\{p\upharpoonright\alpha:\alpha<\kappa^+\}$. By $2^\kappa=\kappa^+$ we have a bijection from partial maps from $\kappa^+$ to $2$ with bounded domain, and $\kappa^+$ itself; so we may replace each $A_p$ with a corresponding $B_p\subseteq \kappa^+$. The collection $\{B_p: p\in 2^{\kappa^+}\}$ is a collection of subsets of subsets of $\kappa^+$, with cardinality $2^{\kappa^+}$, with pairwise intersections of sizes $\le\kappa$.

So in order to have any hope for your principle to fail, we would need to work in a universe where GCH fails everywhere; and the global failure of GCH has large cardinal strength.