# confused about generalized ramsey numbers

The Ramsey number is stated, where $$R(2,3)$$ means Ramsey number where there exists a monochromatic clique of size 2 or a monochromatic clique of size 3.

I also came to understand that $$R(2,3) = N(2,3;2)$$

What does $$N(2,3;3)$$ mean? Especially I do not understand what the 3 means after ;

Regarding this I came across the previous post: Ramsey Number R(4,4), that one user described as:

$$N(m,n;k)$$ is the generalized Ramsey's number $$R(m,n)$$ , except for k colors.

Can someone shine some light on what multicolor ramsey notation $$N(m,n;k)$$ actually means?

• See that user's subsequent comment. – Angina Seng Apr 5 '19 at 6:52
• @LordSharktheUnknown by "the size of colored subsets" does he mean monochromatic $K_k$? So confused. – hiam Apr 5 '19 at 6:56

The notation $$N(m,n,k)$$ does not refer to generalized Ramsey number.
The notation $$N(q_1,\ldots,q_s;r)$$ refers to a specific case, where you start with a set $$S$$ of size $$n$$, and all ($$\binom{n}{r}$$-many) $$r$$-subsets of $$S$$ are split into $$s$$ mutually exclusive families (i.e. $$s$$ colors). Then a theorem states if $$n \geq N(q_1, q_2,\ldots,q_s; r)$$ there is an $$i\leq s$$, and some $$q_i$$-subset of $$S$$ for which every $$r$$-subset is monochromatic.
With $$r=2$$, this is just looking at a coloring of the edges. Hence an equivalence with Ramsey number. As soon as you have $$r>2$$, you are looking at complete $$r$$-uniform hypergraphs , and coloring of their multi-edges.
Generalized Ramsey number are in the form $$R(q_1,q_2,\ldots,q_k)$$ and means that for any $$k$$-coloring of $$K_n$$ with $$n\geq R(q_1,q_2,\ldots,q_k)$$ there exist a monochromatic $$K_{q_i}$$ for some $$i$$.