# How do I do this counting problem? [duplicate]

There are 17 mathematicians and 3 official languages. Every pair of mathematicians communicate in one of the official languages. Prove that there are 3 mathematicians communicating in the same language, pairwise.

I guess pigeonhole principle must be applied but I cannot see how to move forward with it. So, let us consider the three languages (say, A, B and C) as pigeonholes. By pigeonhole principle, at least one of A, B and C must contain at least ceil(17/3) = 6 pigeons. So, 6 mathematicians must be there and all of them know the same language. I can only see this.

• Can’t figure out the pair wise thing. It seems to me that if 6 people all know the same language, they can obviously communicate pairwise. But that can’t be it? Commented Jan 4, 2021 at 14:11
• @DavidMitra This is Ramsey theory. Each language is a different coloured edge. See duplicate. Commented Jan 4, 2021 at 14:30

By PHP, there will be at least $$6$$ mathematicians such that 'A' talks with all the mathematicians in the same language(Language L).