MathDB

Let

n

be an integer greater than

1

and let

X

be an

n

-element set. A non-empty collection of subsets

A_1, ..., A_k

X

is tight if the union

A_1 \cup \cdots \cup A_k

is a proper subset of

X

and no element of

X

lies in exactly one of the

A_i

s. Find the largest cardinality of a collection of proper non-empty subsets of

X

, no non-empty subcollection of which is tight.
Note. A subset

A

X

is proper if

A\neq X

. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

Problem Statement