Finite colourings of $k$-element subsets, monochromatic sets, and the arrow notations $N\to (s,t{)}^2$ and $N\to (r{)}_c^k$

For a set $X$ and a positive natural number $k$, write $[X{]}^k$ for the set of all $k$-element subsets of $X$, where finite cardinality is understood as in The cardinality $\mid A\mid$ of a finite set. A $c$-colouring of $[X{]}^k$ is a function $d:[X{]}^k\to C$ into a set $C$ with $c$ elements. A set $H\subseteq X$ is monochromatic when $d$ is constant on $[H{]}^k$. These notions are unchanged when $X$ is replaced by an equinumerous set (Equinumerous sets, $A\approx B$ and $A⪯B$).

For positive naturals $N,s,t$, the asymmetric arrow

$N\to (s,t{)}^2$

means that every red-blue colouring of the pairs from any $N$-element set has either a red $s$-element set or a blue $t$-element set. Equivalently, the red pairs form a complete graph on some $s$ vertices or the blue pairs form a complete graph on some $t$ vertices. Thus a red-blue colouring witnesses $N\to (s,t{)}^2$ when it contains a red $s$-set or a blue $t$-set.

For positive naturals $N,r,k,c$, the uniform arrow

$N\to (r{)}_c^k$

means that every $c$-colouring of the $k$-element subsets of an $N$-element set has a monochromatic $r$-element set. Natural-number parameters use The natural numbers $\mathbb{N}$ (von Neumann); in particular, all four parameters in this notation are explicitly positive.