confused about generalized ramsey numbers

Question

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?

Answer 1

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.

See Thereoem 3.3 p28 of this Course for details.

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 $i$.