Ramsey Number

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Pigeonhole Principle

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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

m,n	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	1	2	3	4	5	6	7	8	9	10
3	1	3	6	9	14	18	23	28	36	40-43
4	1	4	9	18	25	35-41	49-61	56-84	73-115	92-149
5	1	5	14	25	43-49	58-87	80-143	101-216	125-316	143-442
6	1	6	18	35-41	58-87	102-165	113-298	127-495	169-780	179-1171
7	1	7	23	49-61	80-143	113-298	205-540	216-1031	233-1713	289-2826
8	1	8	28	56-84	101-216	127-495	216-1031	282-1870	317-3583	317-6090
9	1	9	36	73-115	125-316	169-780	233-1713	317-3583	565-6588	580-12677
10	1	10	40-43	92-149	143-442	179-1171	289-2826	317-6090	580-12677	798-23556

References

[1] Weisstein, Eric W. Ramsey Number. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/RamseyNumber.html.