Describe the minimum number of vertices in a simple graph (no loops or multiple edges) with $e$ edges.
I can't derive a formula for this, I don't even know if this is possible.
Anyway, it's not hard to see that the complete graph is the one that optimizes the use of vertices, for example the minimum for $e=3$ is the number of vertices en $K_3$, the minimum for $e=6$ is the number of vertices in $K_4$, the minimum for $e=10$ is the number of vertices in $K_5$ and so on...so I would know this:
If $2 \leq e \leq 3$ the minimum is $3$
If $4 \leq e \leq 6$ the minimum is $4$
If $7 \leq e \leq 10$ the minimum is $5$ and so on...
But I can't see how to derive a formula...is that possible? If so, how?