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Consider the set family $F$ of all $t$-element subsets of $[n]$, for some positive integer $n$.

  • Let $P_1$ be a partition of $F$ into $k$ blocks.
  • Let $P_2 \ne P_1$ be another partition of $F$ into $k$ blocks.

Questions:

  1. Is there anything known about how small $k$ needs to be such that there are at least two $t$-element subsets $S_1,S_2 \in F$ such that there exist blocks $B_1 \in P_1$ and $B_2 \in P_2$ with the property that $S_1,S_2 \in B_1 \cap B_2$.

  2. If the general case is open, is there anything known for more specific cases, e.g., when $t\approx \sqrt{n}$?

  3. Are there any related problems that I could look at?

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  • $\begingroup$ What do you mean by $[n]$? $\endgroup$
    – Armando j18eos
    Commented Dec 29, 2023 at 15:40
  • $\begingroup$ The set $\{1,\dots,n\}.$ $\endgroup$
    – SetFamilyStudent
    Commented Dec 29, 2023 at 17:06

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