MathDB

The Facebook group Olympiad training has at least five members. There is a certain integer

k

with following property: for each

k

-tuple of members there is at least one member of this

k

-tuple friends with each of the other

k - 1

. (Friendship is mutual: if

A

is friends with

B

, then also

B

is friends with

A

.) (a) Suppose

k = 4

. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose

k = 5

. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

Problem Statement