My research interests primarily include Combinatorics and Graph Theory, though I’m not quite sure what I specifically want to study yet. One particular aspect of graph theory that has interested me lately is Ramsey Theory. Ramsey Theory describes certain conditions under which order must appear mathematically. In all of these instances, we find the smallest number of elements so that a certain property will be guaranteed to occur.
For example, if you have 6 people in a room, you can mathematically prove that you must have either 3 who are all mutually friends with each other or 3 who are all mutually strangers. Even more interesting, 6 is the smallest number where this happens (I can find a group of 5 people where it’s impossible to find 3 who are all strangers or 3 who are all friends; try to find such a group among your friends!), and this is how the 3rd Ramsey number is defined.
What if we wanted to guarantee that there are 4 people who are all friends or 4 people who are all strangers? Turns out you need 18 people. It gets really interesting when you ask the question for 5 people (or more); no one knows the answer! We have bounds for what it must be, but that’s about it. Erdos, the famous mathematician, had a quote, “Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find [the fifth Ramsey number]. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded [the sixth Ramsey number], however, we would have no choice but to launch a preemptive attack.”
Interesting concepts like these are just few of the many available to combinatorics and graph theory, and they’re what I like to spend my time pondering.