We are given a collection of sets $A_1,\ldots,A_s$, pairwise different and each of cardinality $k$, and a collection of sets $B_1,\ldots,B_s$, pairwise different and each of cardinality $l>k+1$, such that $A_i\subseteq B_i$ for all $i=1,\ldots,s$. Can we find elements $a_i$ from $B_i\setminus A_i$ for all $i=1,\ldots,s$ such that the sets $A_i \cup \{a_i\}$ are all different?
Rephrasing this in terms of the corresponding subset lattice, this question is asking whether there are $s$ vertex-disjoint monotone paths between the sets $A_i$ and $B_i$, $i=1,\ldots,s$, in this graph.
I would be interested in knowing the answer to this question, or learning about related questions or references. Many thanks!
[Edit: It has been observed by Bjørn Kjos-Hanssen that the special case $l-k\geq s$ (="very few sets") is trivially true. Similarly, the special case $l-k\geq k+1$ (="set sizes differ a lot") can be easily proved by Hall's theorem. The first special case that falls outside these trivial ranges and that might therefore be interesting to consider is $k=2$ and $l=4$ (and $s\gg 1$).]