A bound is tight if it can be realized. A bound that is not tight is sometimes said to be slack.
For example, let be the collection of all finite subsets of in general position. Define a function as follows. First, let the weight of an element of be the size of its largest convex subset, that is,
The function is defined by
that is, is the smallest number such that any collection of points in general position contains a convex -gon. (By the Erdős-Szekeres theorem (http://planetmath.org/HappyEndingProblem), is always finite, so is a well-defined function.) The bounds for due to Erdős and Szekeres are
The lower bound is tight because for each , there is a set of points in general position which contains no convex -gon. On the other hand, the upper bound is believed to be slack. In fact, according to the Erdős-Szekeres conjecture, the formula for is exactly the lower bound: .