Number of subsets of $\mathfrak c$ that are different no matter how high you go? I am interested in collections $\mathcal X \subseteq \mathcal P(\mathfrak c)$ such that for any distinct $X,Y\in\mathcal X$ and $\alpha<\mathfrak c$ we have $X\cap [\alpha,\mathfrak c]\neq Y\cap [\alpha,c]$.
How big can such a collection be? $2^\mathfrak c$? Does it depend on the cofinality of $\mathfrak c$?

From $\mathfrak c^2=\mathfrak c$ it follows that $\mathfrak c$ is the union of a collection of $\mathfrak c$ pairwise disjoint subsets, each of cardinality $\mathfrak c.$ This family has $2^\mathfrak c$ different subfamilies, and the unions of two different subfamilies will be subsets of $\mathfrak c$ that are "different no matter how high you go".
