The research about Ramsey Numbers.

In combinatorial mathematics, Ramsey's theorem states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let r and s be any two positive integers. Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. (Here R(r, s) signifies an integer that depends on both r and s.)

Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of monochromatic subsets, that is, subsets of connected edges of just one colour.

An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours, c, and any given integers n1, …, nc, there is a number, R(n1, …, nc), such that if the edges of a complete graph of order R(n1, ..., nc) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all colour i. The special case above has c = 2 (and n1 = r and n2 = s).

The Ramsey number R(m,n) gives the solution to the party problem, which asks the minimum number of guests R(m,n) that must be invited so that at least m will know each other or at least n will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices v=R(m,n) such that all undirected simple graphs of order v contain a clique of order m or an independent set of order n. Ramsey's theorem states that such a number exists for all m and n.

Ramsey Number

============> 1, 2, 3

• :::::>1, 2
• :::::>1, 3
• :::::>2, 3

============> 1, 2, 4

• :::::>1, 2
• :::::>1, 4
• :::::>2, 4

============> 1, 2, 5

• :::::>1, 2
• :::::>1, 5
• :::::>2, 5

============> 1, 3, 4

• :::::>1, 3
• :::::>1, 4
• :::::>3, 4

============> 1, 3, 5

• :::::>1, 3
• :::::>1, 5
• :::::>3, 5

============> 1, 4, 5

• :::::>1, 4
• :::::>1, 5
• :::::>4, 5

============> 2, 3, 4

• :::::>2, 3
• :::::>2, 4
• :::::>3, 4

============> 2, 3, 5

• :::::>2, 3
• :::::>2, 5
• :::::>3, 5

============> 2, 4, 5

• :::::>2, 4
• :::::>2, 5
• :::::>4, 5

============> 3, 4, 5

• :::::>3, 4
• :::::>3, 5
• :::::>4, 5