What does it mean when a bound is sharp?

Question

I've been reading this paper about the maximum size of digraphs. The author mentions that one of the digraph has a sharp bound. What does it mean when a bound is sharp?

Answer 1

Let's suppose that $X$ is suitably closed, otherwise we have to work with suprema. We say $y$ is a sharp upper bound for $X$ if $y\geq x$ for all $x \in X$ and there exists an $x_0 \in X$ such that $y=x_0$.

That is, $y$ is the best possible upper bound that can be found.

Answer 2

As an example, let $A=[0,1]$. Then $1$ is a sharp upper bound for the elements in $A$, since $1\ge \text{all } a\in A$. The numbers $2$ or $3$ or any number $>1$ are also upper bounds, however not sharp. A sharp bound cannot be improved upon, since there must at least one case where the condition holds with equality.