probabilistic method

概率方法

Combinatorics · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: probabilistic existence method

Translation evidence: Literature usage

Meaning in context

The probabilistic method proves existence of a combinatorial object by defining a distribution on candidates and showing that the desired property has positive probability, sometimes after a controlled alteration.

概率方法通过在候选组合对象上定义分布，并证明所需性质有正概率出现来证明存在性；有时还需进行受控的修改。

Example

If a nonnegative integer-valued bad-configuration count X has EX<1\mathbb E X<1, some outcome has X=0X=0.

Usage and distinctions

A sufficient expectation bound may fail even when a good object exists; an existence proof is not automatically an efficient construction algorithm.

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Definition sources

  1. Keller and Trotter — Applied Combinatorics (2016)Section 11.6, Theorem 11.7, random-graph construction and alteration proof · Checked 2026-10-14

Translation sources

  1. Reinhard Diestel — 图论，第五版，官方中文目录目录，§11.2「概率方法」（名称证据） · Checked 2026-10-14
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