I think the Erdős-Szekeres conjecture (see the Wikipedia article on the Happy Ending Problem), which says that the smallest $N$ for which any $N$ points in general position in the plane contain the vertices of some convex $n$-gon is $N=2^{n-2}+1$, is a little bit ridiculous in that there is not a lot of reason to believe it. Indeed, only the cases $n=3,4,5,6$ are known, and while it is known that $2^{n-2}+1$ is a lower bound for this $N$, I really am not aware of any other substantial evidence in favor of this conjecture other than that it's a very nice pattern.