MathDB

Let

A

be a finite set of points in the coordinate plane. Suppose that

A

has

n\geq3

points. Given any

a

A

, the horizontal and vertical lines through

a

define four closed quadrants centered at

a

. For any real number

\alpha

, call a point

a

A

\alpha

-good if there are two diagonally opposite closed quadrants centered at

a

that each contain at least

\alpha n

points from

A

. Show that there is some

a

A

that is

\frac{1}{8}

-good.

Problem Statement