This theorem is named after frank ramsey. 1930, he proved that r (3,3) = 6 in A Problem in Formal Logic. The definition of Ramsey number has two descriptions in the language of graph theory: for all N-vertex graphs, there is a group of K vertices or an independent set of L vertices. The smallest natural number n with this property is called Ramsey number, which is denoted as r (k, l); In coloring theory, it is described as follows: For any two edges of a complete graph Kn are colored (e 1, e2) so that Kn[e 1] contains a sub-complete graph of order K and Kn[e2] contains a sub-complete graph of order L, then the minimum n satisfying this condition is called a Ramsey number. (Note: Ki represents a complete graph of order I according to the notation of graph theory) Ramsey proved that the answer to a given positive integer k and l, R(k, l) is unique and limited. Ramsey number can also be extended to more than two numbers: each edge of the complete graph Kn is painted with one of R colors, which are respectively expressed as e 1, e2, e3, ..., er, respectively. In Kn, there must be an l 1 sub-complete graph with color e 1, or an l2 sub-complete graph with color e2 ... or a partially complete graph with color E2. Meet the requirements and the least number n is R(l 1, l2, l3, ..., lr; R). The number of Ramsey numbers or the number of Ramsey numbers whose upper and lower bounds are known is very small. Paul Edith once described the difficulty of finding Ramsey number with a story: "Imagine an alien army landing on the earth and asking for the value of r (5,5), otherwise it will destroy the earth. In this case, we should concentrate all computers and mathematicians to try to find this value. If they demand the value of R (6,6), we will try to destroy these aliens. " Obvious formulas: R( 1, s)= 1, R(2, s)=s, R(l 1, l2, l3, ..., LR; r)=R(l2，l 1，l3，...，lr； r)=R(l3，l 1，l2，...，lr； R) (Changing the order of Lee does not change the value of Ramsey). r， s 3 4 5 6 7 8 9 103 6 9 14 18 23 28 36 40–434 9 18 25 35–4 1 49–6 1 56–84 73– 15 92– 1 49 5 14 1 17 17 23 49 – 6 1 80 – 143 1 13 – 298 205 – 540 2 16 – 103 / kloc-0/ 233 – 17 13 289 – 28268 28 56 – 84 10 1 – 2 16 127 – 495 2 16 – 654 38+003 1 282 – 1870 3 17 – 3583 3 17 – 60909 36 73 – 1 15 125 – 3 16 169 – 780 233 – 17 / kloc-0/3 3 17 – 3583 565 – 6588 580 – 12677 10 40 – 43 92 – 149 143 – 442 179 – 1 17 / The proof that kloc-0/289–2826 317–6090580–1267798–23556r (3,3) =17r (3,3) is equal to 6 is in a complete graph of K6, if Choose any endpoint p, which has five edges connected with other endpoints. According to the principle of pigeon's nest, at least two of the three faces have the same color, and it is generally assumed that this color is red. On the three endpoints of these three sides except P, three sides are connected with each other. If any of these three sides is red, the two endpoints of this side and the two sides connected with P form a red triangle. If any of these three sides are not red, they must be blue, so they form a blue triangle. In K5, there is not necessarily a red triangle or a blue triangle. The connection between each endpoint and two adjacent endpoints is red, and the connection between each endpoint and the other two endpoints is blue. The popular version of this theorem is the friendship theorem.