MathDB

Consider any rectangular table having finitely many rows and columns, with a real number

a(r, c)

in the cell in row

r

and column

c

. A pair

(R, C)

, where

R

is a set of rows and

C

a set of columns, is called a saddle pair if the following two conditions are satisfied:
[*]

(i)

For each row

r^{\prime}

, there is

r \in R

such that

a(r, c) \geqslant a\left(r^{\prime}, c\right)

for all

c \in C

; [*]

(ii)

For each column

c^{\prime}

, there is

c \in C

such that

a(r, c) \leqslant a\left(r, c^{\prime}\right)

for all

r \in R

.
A saddle pair

(R, C)

is called a minimal pair if for each saddle pair

\left(R^{\prime}, C^{\prime}\right)

with

R^{\prime} \subseteq R

and

C^{\prime} \subseteq C

, we have

R^{\prime}=R

and

C^{\prime}=C

. Prove that any two minimal pairs contain the same number of rows.

Problem Statement