# At least ten language is spoken

At a party of $250$ mathematician , each mathematician speaks one or more languages. It is found that for any two mathematicians, each speaks at least one language not spoken by other. Show that there are at least $10$ different languages spoken in the party.

I have a feeling that Pigeon Hole Principle to be used, but I am clueless how to use it. Please help.

• Do you know Sperner's theorem? – Mirko Jan 31 '17 at 10:11