NL-BXMO-TST 2016 Problem 4

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?