How to find the Ramsey numbers?I am new in graph theory and I need help. By PHP,I have proved that $R(3,3)$=6.But I am finding difficulty when the numbers get bigger. Is their any particular method of finding it.
For example : Please show mw how to find $R(4,3)$ .
No, there is no known method of finding Ramsey numbers (Edit: No efficient known method; i.e brute-force bash and 'clever' brute force bashes are the best algorithms so far, and even these are too slow to compute e.g $R(5,5)$ or $R(6,6)$ afaik; see the wikipedia article for more). In general, the best you can do is bound them. For instance, $R(r,s)\leq R(r-1,s)+R(r,s-1)$ allows you to say that $R(4,3)\le R(3,3)+R(4,2)=10$ is one such bound.
To say that $R(r,s)=n$, one must provide a colouring of $K_{n-1}$ that works (which is often the easier part), and also prove that any colouring of $K_n$ must contain a clique. No known method is known for either of the two parts. For the specific case of $R(4,3)$,
is an example of why $R(4,3)>8$ -- this colouring (with the complementary edges coloured red) has no blue complete graphs on $4$ edges, nor any complete red graphs on $3$ edges.
To show why $R(4,3)\le 9$, note that if you have $9$ vertices, and a colouring, consider any vertex, $V$. If the red degree is $4$ or higher, these four either form a blue clique or a red triangle with $V$. Thus the red degree of any vertex is at most $3$ so the blue degree is at least $5$. But it is well known that any five vertices have a triangle and thus the theorem is proven.