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单词 Tight
释义

tight


A bound is tight if it can be realized. A bound that is nottight is sometimes said to be slack.

For example, let 𝒮 be the collection of all finite subsetsof 2 in general positionMathworldPlanetmath. Define a functiong: as follows. First, let the weight ofan element S of 𝒮 be the size of its largest convex subset,that is,

w(S)=max{|T|:TS and T is convex}.

The function g is defined by

g(n)=min{|S|:w(S)n},

that is, g(n) is the smallest number such that any collection ofg(n) points in general position contains a convex n-gon. (By theErdős-Szekeres theorem (http://planetmath.org/HappyEndingProblem), g(n) isalways finite, so g is a well-defined function.) The bounds for gdue to Erdős and Szekeres are

2n-2+1g(n)(2n-4n-2)+1.

The lower bound is tight because for each n, there is a set of2n-2 points in general position which contains no convex n-gon.On the other hand, the upper bound is believed to be slack. In fact,according to the Erdős-Szekeres conjecture, the formula for g(n)is exactly the lower bound: g(n)=2n-2+1.

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更新时间:2025/5/4 22:21:55