Math 2.0

Combinatorics Theorem

Probabilistic lower bound for diagonal Ramsey numbers

Builds on Ramsey number

Statement

Let nn be a positive integer. Then the diagonal Ramsey number R(n,n)R(n,n) satisfies

R(n,n)≥ne2 2n/2.R(n,n) \ge \frac{n}{e\sqrt{2}}\, 2^{n/2}.

Remarks

The bound shows that R(n,n)R(n,n) grows at least exponentially in nn, with base 2\sqrt{2}. It is proved by the probabilistic method, which colours the edges at random.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Ramsey number, definition, no evidence levelRamsey numberdefinitionProbabilistic lower bound for diagonal Ramsey numbers, statement, no evidence levelProbabilistic lower b…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Ramsey number definition
  2. Probabilistic lower bound for diagonal Ramsey numbers statement

Further reading

  1. Web https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_ramsey_probability.ptx#L7, source/s_ramsey_probability.ptx:7
About this page
TagSAZNT80CG5
URIm2://SAZNT80CG5
Kindstatement, theorem
WritingW0 stub
MSC05D10, 05C55, 05D40
StatusThis copy of the site has no build data. Run m2 status SAZNT80CG5.
AboutRamsey number
UsesRamsey number
Changed4 November 2026

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