MathDB

Given a set of boys and girls, we call a pair

(A,B)

amicable if

A

and

B

are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions:
i) The set has the same number of boys and girls.
ii) For every four different people

A,B,C,D

if the pairs

(A,B),(B,C),(C,D)

and

(D,A)

are all amicable, then at least one of the pairs

(A,C)

and

(B,D)

is also amicable. iii) At least

\frac{1}{2013}

of all boy-girl pairs are amicable.
Let

m

be a positive integer. Prove that there exists an integer

N(m)

such that if a affectionate set has al least

N(m)

people, then there exists

m

boys that are pairwise friends or

m

girls that are pairwise friends.

Problem Statement