4

Part of 1989 Balkan MO

Problems(1)

a super nice and difficult 4-th problem proposed by Romania

Source: bmo 1989

The elements of the set

F

are some subsets of

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

and satisfy the conditions: i) if

A

F

A

has three elements; ii)if

A

B

are distinct elements of

F

A

B

have at most one common element. Let

f(n)

be the greatest possible number of elements of

F

\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}

set theorycombinatorics proposedcombinatorics