Famous Problems Where We Only Know the Elementary Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. For the problem as stated (there are many variations), the best known bounds are $4 \leq \chi \leq 7$. The lower bound comes from the existence of a clever subgraph on just seven vertices that is readily seen to require four colors. The upper bound comes from a straightforward tiling of the plane by monochromatic hexagons that admits a proper $7$-coloring.
I like to show this problem to young students because they are always fascinated by the fact that we can cover "everything" that is known about it in just a few minutes. What are some other problems of this type?

What are some famous problems for which the best known results are fairly obvious or elementary?

Update: Aubrey de Grey has recently improved the lower bound to 5 using an elaborate subgraph, so we now know slightly more than the elementary.

Here's a little ditty from Complexity Theory.
The following facts are quite easy to prove:


*

*$\mathsf{L} \subseteq \mathsf{NL} \subseteq \mathsf{P} \subseteq \mathsf{NP} \subseteq \mathsf{PSPACE} \subseteq \mathsf{EXPTIME} \subseteq \mathsf{EXPSPACE}$;

*$\mathsf{L} \neq \mathsf{PSPACE}$;

*$\mathsf{PSPACE} \neq \mathsf{EXPSPACE}$.


By (2) it immediately follows that at least one of the first four inclusions in (1) is proper,  and, similarly, by (3) at least one of the last two inclusions in (1) is proper.  However it is completely unknown which inclusions in (1) are proper (though it is strongly suspected that each is).  (Note that this subsumes the famous $\mathsf{P} \overset{\mathord{?}}{=} \mathsf{NP}$ problem.)
