Why is $R^{(3)}(s,t)=\max\{s,t\}$,if $\min\{s,t\}=3$
Did I understand something wrong ?
If a graph has $|S|$ vertices, we let $\binom{|S|}{3}$ the set of all subsets of $3$ elements of $S$.
Then we give every such 3-tuple a color.
And $R^{(3)}(s,t)$ is the number $n$ s.t. every coloring of 3-tuples $\binom{n}{3}$, using $2$ colors, say red and blue, gives either a red-homogeneous set of size $s$ or blue homogeneous set of size $t$.
for example red-homogeneous set $S'$ means that, every $3$-tuple formed by the elements of $S'$ receives the color red.
So in our case if $\min\{s,t\}\overset{\text{wlog}} =s=3$, then does this not mean that, we need an $n$ such that every coloring of the $3$-tuples of this set of size $n$ in red and blue has
either a subset of $s=3$ elements s.t. every $3$-tuple in it is red, or
a subset of $t$ elements s.t. every $3$-tuple in it is blue
don't we have $R^{(3)}(s,t)=R^{(3)}(t,s)$? Then we would only need $3$ elements, am I wrong?
EDIT: My Question is from the script below on page $70$, last paragraph before Theorem $12.4$