My research deals with finite discrete structures (as opposed to the continuous structures more common in classical mathematics). These kinds of structures occur frequently in modern application of mathematics.
Quite often I work with graphs, which are sets of points, some pairs of which are associated in some way, which can be represented by an edge joining them. Suppose, at a party, each pair of people present are either friends or enemies. It is possible to have a party of 5 people without either three mutual friends or three mutual enemies, but it is impossible to avoid one of these occurring in a party of 6 (try it).
It is possible to have a party of 17 without four mutual friends/enemies, but not 18. It is known that if the party is big enough you will get five mutual friends/enemies, but how large must it be? No-one knows. Surprisingly, this little puzzle has profound applications.