[sdenton4]

An old favorite open graph theory problem:

In your right hand, take any collection of trees with 2, 3, ..., n vertices. These have a total of 1+2+...+n-1 edges, ie, n choose 2.

In your left hand, take the complete graph with n vertices. This, of course, has the same number of edges.

Conjecture: it is always possible to pack/embed the trees into the complete graph in such a way that all edges are matched exactly once.

The problem has been open for like fifty years, lacking a counter-example or a proof. One can assume Erdos thought hard about it at some point...

[a3_nm]

Interesting! It looks like this is called the "Gyárfás tree packing conjecture".