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Jul 16, 2021 at 7:32 comment added Tony Huynh Yes, such families do not always exist, so this is not a complete answer. It does give an infinite set of parameters for which the obvious upper bound can actually be achieved.
Jul 16, 2021 at 7:28 comment added Nate River Hm, each pair being in at most one $A_k$ may not be possible in the general case, depending on the values of $k, m, n$. Oh though I guess thats what you meant when you talked about existence of such families.
Jul 16, 2021 at 7:24 history answered Tony Huynh CC BY-SA 4.0