I have a problem I'm completely stuck with, but which seems very interesting.
17 mathematicians from different countries are all writing to each other. Every two mathematicians write to each other in one of three languages: English, French or Russian. Prove that there exists 3 mathematicians, who all write to each other in the same language.
Now, I'm fairly new to writing proofs, so I'd appreciate it if you tried to explain it as clearly as you can.
I suspect this has something to do with combinations, but I can't be sure.
Pick a person, $X$, at random and divide the other $16$ up by the language in which they converse with that person. One of those groups must have at least $6$ people in it. Wlog let's say that common language is English.
Now if any two of those $6+$ speak to each other in English we are done, so assume they only use the other two languages between them. Take a person $Y$ from the group and divide the remaining $5+$ up according to the language they speak with $Y$. One of those two groups must have at least $3$ people in it. Let's say those $3+$ all speak Russian with $Y$. If any two of the $3+$ speak Russian to each other, we are done. But if they all speak French to each other we are also done.