Is there anything known about the following problem? Which (sub)field to look into for questions of this flavor?

Consider a collection $F$ of subsets of $[n]$, excluding the empty set, with the property that every element of $[n]$ is contained in exactly $n$ sets. The same set might appear more than once in $F$ (The number of sets in $F$ is thus between $n$ and $n^2$ and the sum of cardinalities is $n^2$).

Is there a sequence (possibly with repetitions) of the numbers $1 \dots n$, of length at most $n^\alpha$, such that each set appears somewhere in the sequence in at most $n^\beta$ contiguous blocks.

Here's a small example: F = { {1,2}, {1,3}, {2,3}, {1,2,3} }. In the sequence (1,2,3) the set {1,3} can be mapped to two blocks only. In the sequence (1,2,3,1) all sets can be mapped to a single block.

$\alpha = 1, \beta=1$ works if we use any permutation of the numbers.

$\alpha = 2, \beta =0$ works if we concatenate all sets of F.

Can we get something "in between" ?

EDIT: as an important special case, F could consist of n "partitions" of [n].