Ramsey problem : $R(p, q, 3) \leq R(p, q) + R(p-1, q) + R(p, q-1) - 1$ Given $p, q, r \in \mathbb{N}$, let $R(p, q)$ denote the smallest natural number "$n$" such that for any coloring of edges of an $n$-clique by 2 colors : blue or red, there exists either a "blue $p$-clique" or a "red $q$-clique".
Let $R(p, q, r)$ denote the smallest natural number "$n$" such that for any coloring of edges of an $n$-clique by 3 colors : blue, red or yellow, there exists either a "blue $p$-clique", "red $p$-clique" or a "yellow $q$-clique".
Prove that for all integers $p, q \geq 2$, $R(p, q, 3) \leq R(p, q) + R(p-1, q) + R(p, q-1) - 1$.
Please give me some suggestions.
 A: You are familiar with the proof of
$$R(p,q)\ \le\ R(p-1,q)+R(p,q-1)\tag1$$
which can also be written in the form
$$R(p,q)-1\ \le\ 1+[R(p-1,q)-1]+[R(p,q-1)-1]\tag2$$
which is a good way to think about it, seeing as $R(p,q)-1$ is the largest $n$ for which an $n$-clique can be colored blue & red without making a blue $p$-clique or a red $q$-clique.
The inequality (2) can be generalized (with essentially the same proof) to any number of colors; e.g., for three colors
$$R(p,q,r)-1\ \le\ 1+[R(p-1,q,r)-1]+[R(p,q-1,r)-1]+[R(p,q,r-1)-1]\tag3$$
(see proof below), which simplifies to
$$R(p,q,r)\ \le\ R(p-1,q,r)+R(p,q-1,r)+R(p,q,r-1)-1.\tag4$$
Setting $r=3$ and noting that $R(p,q,2)=R(p,q)$ we get
$$R(p,q,3)\ \le\ R(p-1,q,3)+R(p,q-1,3)+R(p,q)-1\tag5$$
which somewhat resembles your inequality
$$R(p,q,3)\ \le\ R(p-1,q)+R(p,q-1)+R(p,q)-1.\tag{6}$$
But (6) is incorrect.
Example. For $p=2,q=3,$ the inequality (6) says that
$$R(2,3,3)\ \le\ R(1,3)+R(2,2)+R(2,3)-1=1+2+3-1=5$$
but in fact $R(2,3,3)=R(3,3)=6.$
Example. For $p=q=3$ the inequality (6) says that
$$R(3,3,3)\ \le\ R(2,3)+R(3,2)+R(3,3)-1=3+3+6-1=11$$
but in fact $R(3,3,3)=17=R(2,3,3)+R(3,2,3)+R(3,3,2)-1.$
Looks like there was a typo in the assignment.
Proof of (3). Let $n=R(p,q,r)-1.$ Color the edges of an $n$-clique with three colors (blue, red, yellow) so that there is no blue $p$-clique, no red $q$-clique, and no yellow $r$-clique. Choose a vertex $v$ and suppose that $v$ is joined to $n_1$ vertices by blue edges, to $n_2$ vertices by red edges, and to $n_3$ vertices by yellow edges. Then $R(p,q,r)-1=n=1+n_1+n_2+n_3$
$\le\ 1+[R(p-1,q,r)-1]+[R(p,q-1,r)-1]+[R(p,q,r-1)-1].$
