What are some theorems that are intuitively true but very difficult to prove?
Today I came across the Jordan curve theorem, which seems as if it should be very easy to prove.
The basic statement is this: given any closed, non-self-intersecting loop, it is impossible to draw a path from a point outside the loop to a point inside the loop without crossing the edge of the loop. Here is an example.
This fact went unproven for a very long time, and the current proofs are incredibly long. The reason is that loops can be much more complex than anything that can be drawn by humans. They can contain fractal edges and be nowhere-differentiable.
Here is an example of a proof for the Jordan curve theorem. It is a computer-assisted proof, so it cannot be understood unless you know the language; however, its length demonstrates how hard it is to prove the Jordan curve theorem.
My question is: what are some other examples of mathematical facts that are very intuitively true, but which are very difficult to actually prove?