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Family of sets

Source: 2024 Taiwan TST Round 1 Independent Study 2-C

March 7, 2024

combinatoricsTaiwan

Problem Statement

A

k

-set is a set with exactly

k

elements. For a

6

-set

A

and any collection

\mathcal{F}

of

4

-sets, we say that

A

is

\mathcal{F}

-good if there are exactly three elements

B_1, B_2, B_3

in

\mathcal{F}

that are subsets of

A

, and they furthermore satisfy

(A \backslash B_1) \cup (A \backslash B_2) \cup (A \backslash B_3) = A.

Find all

n \geq 6

so that there exists a collection

\mathcal{F}

of

4

-subsets of

\{1, 2, \ldots , n\}

such that every

6

-set

A \subseteq \{1, 2, \ldots , n\}

is

\mathcal{F}

-good.
Proposed by usjl

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