# Difference between revisions of "Ramsey Number"

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## Definition

Ramsey number $R(m, n)$ is the solution to the party problems, which ask the minimum number of guests that must be invited so that at least $m$ will know each other or at least $n$ will not know each other.

## A Summary of Known Ramsey Numbers

 r,s 1 2 3 4 5 6 7 8 9 10 1 1 1 1 1 1 1 1 1 1 1
 m, n 1 2 3 4 5 6 7 8 9 $i=2$ $10^7(1-\frac{1}{10^7})=9999999$ $i=3$ $10^7(1-\frac{1}{10^7})^2=9999998.0000001$ $i=4$ $10^7(1-\frac{1}{10^7})^3=9999997.00000029999999 \approx 9999997.0000003$ $i=5$ $10^7(1-\frac{1}{10^7})^4=9999996.000000599999960000001 \approx 9999996.0000006$ $i=...$ $...$ Image X[1] $i=101$ $10^7(1-\frac{1}{10^7})^{100} \approx 9999900.00049505$

## Examples

1. Napier, 1616, p. 46