MathDB

A

is a subset of

X

, then we take

A^1 = A

A^{-1} = X - A

. The subsets

A_1, A_2, \ldots, A_k

are called mutually independent if the product

A_1^{\varepsilon_1} \cap A_2^{\varepsilon_2} \ldots A_k^{\varepsilon_k}

is nonempty for every system of numbers

\varepsilon_1 , \varepsilon_2, \ldots, \varepsilon_k

, such that

|\varepsilon_2| =

1 for

i = 1, 2, \ldots, k

. What is the maximum number of mutually independent subsets of a

2^n

-element set?

6