ANDREW THOMASON Mathematics
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.