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