MathDB

Let

S

be a finite set of points in the plane. A linear partition of

S

is an unordered pair

\{A,B\}

of subsets of

S

such that

A\cup B=S,\ A\cap B=\emptyset,

and

A

and

B

lie on opposite sides of some straight line disjoint from

S

(

A

B

may be empty). Let

L_{S}

be the number of linear partitions of

S.

For each positive integer

n,

find the maximum of

L_{S}

over all sets

S

n

points.

B3