This question arises from a request for an algorithm to do such, from 9 sets of 12 elements, arrange 12 groups of the 9 elements, selecting 1 element from each set

Given a set, $S$, of sets, $S_i$, $\mid S\mid = s$ where:

$\forall S_i \in S, \mid S_i\mid = e$, all sets in $S$ have the same cardinality, $e$, and $e > 1$

$\forall S_i, S_j \in S, S_i \cap S_j = \phi, i \ne j$, all sets in $S$ are pairwise disjoint

By choosing one element from each $S_i \in S$ create a new set $C_k$, and hence $\mid C_k\mid = s$.

Let $P_k = \{ \{c_p, c_q\} : c_p, c_q \in C_k, p \ne q \}$, $P$ is the set of binary subsets of $C$.

Do there exist $C_k, k = 1$ to $se$ such that $l, m = 1$ to $se$, $P_l \cap P_m = \phi, l \ne m$?

My conjecture is that for the $se$ sets, $C_k$, to exist, then necessarily $s \le e$ and $e$ is prime.

How might this be proved, or disproved (if there is no readily findable counter example)?

Apologies in advance if this is a known result. If so, please provide a reference. (It has been a very long time since I have done any formal math and I am a very bit "rusty".)

[EDIT - CORRECTION] $e$ needs to be prime, not $s$ - corrected above

`0123 4567 89AB CDEF`

and e.g. $C_k$ as`048C 059D 06AE 07BF 149F 158E 16BD 17AC 24AD 25BC 268F 279E 34BE 35AF 369C 378D`

. $\endgroup$4more comments