So I went through the definition of Ramsey number and I have a basic question.
Definition: For any given number of colours, $c$, and any given integers $n_1, …, n_c$, there is a number, $R(n_1, …, n_c)$, such that if the edges of a complete graph of order $R(n_1, ..., n_c)$ 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.
Question: Is the multicolour Ramsey number $R(n_1,n_2,..n_c) $same as $R(n_2,n_1,..n_c) $or any other permutation of$ {n_1,n_2,..n_c}$? The definition seems to imply so, I just want to verify if I'm thinking right. I have read that $R(m,n)=R(n,m)$ but nothing about symmetricity of multicolour Ramsey numbers.
Yes. It is just a matter of using different colours. For example, In $R(3,4,5)$ we want either a red $K_3$, a blue $K_4$ or a green $K_5$ whereas in $R(5,3,4)$ we want either a red $K_5$, a blue $K_3$ or a green $K_4$, which doesn't make a difference at all.