Problem

In a mathematical competition some competitors are friends. Friendship

is always mutual. Call a group of competitors a clique if each two of them are friends. (In

particular, any group of fewer than two competitors is a clique.) The number of members

of a clique is called its size.

Given that, in this competition, the largest size of a clique is even, prove that the

competitors can be arranged in two rooms such that the largest size of a clique contained

in one room is the same as the largest size of a clique contained in the other room.

Solution

Let the largest clique C have the size 2n. We can divide this clique in two parts and send one part to room A and another part to room B. Let’s call these cliques CA and CB. The rest of the participants will go to room A. In room A there might be a clique V of the size s>n. There are participants in clique V that do not belong to clique CB (If it’s not the case then send the rest of the participants to room B and apply the same reasoning. This subset of the participants can’t belong to both CA and CB). Send one participant from room A belonging to clique L to room B. The size of L decreased by one. The size of the largest clique in B remains unchanged. Proceed till the size of L will become n+1. Send one more participant to B. The size of the largest clique in B will not exceed n since the size of L was less or equal to 2n. Therefore in both rooms the largest clique will have the size n. QED.

]]>I do not know who you are talking about.

]]>in the mid-1970s Moscow Volodya Grinberg was a star of every maths competition I went to.

He didn’t make it to Moscow University because of anti-Semitism, but is now a full maths professor at UCLA

Do you know if he is related to Natalia?

Regards

Faibsz

]]>Israel Gelfand lived in Moscow for many years, so the chess player can’t be a close relative. Otherwise, I do not know.

]]>If I may ask a somewhat indiscreet question (?) does Prof. Gelfand has any relation to the world famous chess player Boris Gelfand from Bellarussia? (now representing Israel at international chess competitions) ]]>