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?
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5In $a\le b$, $b$ is an upper bound. It is said sharp if the equality $a=b$ indeed occurs, so that you cannot find a lower upper bound. For instance, $\sin x\le1$ is sharp; $\sin x\le2$ is true but not sharp. – Oct 26 '16 at 10:03
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2Just one addition, https://en.wikipedia.org/wiki/Upper_and_lower_bounds#Tight_bounds The bound is called sharp if equality holds for at least one value of x; The bound is called tight if no smaller/larger value is a bound. See also https://cs.stackexchange.com/questions/19141/what-is-an-asymptotically-tight-upper-bound – Henry.L Jan 10 '21 at 01:30
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Does this answer your question? "Sharp" Inequalities – Pietro Paparella Sep 11 '23 at 21:51
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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.
Dan Rust
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1There's no generlisation about it. A supremum of a set is a sharp upper bound of that set. – Dan Rust Oct 01 '19 at 16:45
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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.
Jimmy R.
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