There is a particular problem in a branch of mathematics called Ramsey theory. The problem is somewhat difficult to explain clearly as it involves something called a hypercube, which is just your basic cube (well, square, really) generalized to n dimensions. This is a 4-cube (a 4 dimensional hypercube) projected into three-dimensional space (then projected into 2 dimensional space, since you’re looking at it on a 2 dimensional computer screen):
Anyway, that’s not really the hilarious part. The Ramsey theory problem is to find the smallest number of dimensions for which connecting each vertex of an n-cube by a line segment and then colouring each line segment will force you to have a single-coloured 4-vertex planar complete subgraph. Don’t worry if you didn’t follow that; you’ll still understand the funny part. But just to give you an idea of the problem, this is Wikipedia’s example of what the problem means with a 3-cube (the regular type of cube):
The thing under the cube is a single-coloured 4-vertex planar complete subgraph—it’s just a square with an X inside of it that is all one colour (don’t worry, the funny part is coming). So again, the problem is to try to figure out what is the smallest dimension N of the hypercube for which it is impossible to not have a single-coloured square with an X in it. It turns out that it’s possible to construct such a 3-cube, so N>3.
So the problem isn’t solved, per se, but in 1971 a fellow named Ronald Graham determined an upper bound on the solution. The upper bound is known as Graham’s number, and it is so large that you cannot express it with regular notation. There is literally not enough space in the universe; if you tried to write it out in a normal digital representation (like 4938293… etc), and each digit took up the smallest amount of space that we know how to think about, the representation would take up more space than there is in the universe. Incidentally, we somehow know how to calculate the last digits of Graham’s number—they are 2464195387.
So that’s the upper bound on the solution. There is also a lower bound. Would you like to know what the lower bound is? It’s 13. So, to sum up, the solution to the above problem is known to exist. Let’s call it N. Here’s what mathematics has been able to prove about N:
13 ≤ N ≤ some number that is considerably larger than the number of atoms in the Universe.
This is what mathematicians do for a living.