I have found proof that the Ramsey number $R(p, q; r)$ exists by induction in three variables, but it is hard for me to understand this induction. The definition of ramsey number requires that $p,q \geq r $
Let's see if someone can clarify this.
Fist prove that $ R (p, q; 1) = p + q-1 $, $ R (r, q; r) = q $ and $ R (p, r; r) = p $. This seems to be the base case of the induction principle.
Then, suppose that
$i)$ $R (p, q; r - 1)$ exists for all $p, q$.
$ii)$ There is the number $q1 = R (p, q - 1; r)$ for all $p, r$.
$iii)$ There is the number $p1 = R (p - 1, q; r)$ for all $q, r$.
and proof that it is also true for $ p, q, r $
Finally he assume that $ R (p, q; r) $ does not exist, and uses items $i, ii$ and $iii$ at the same time to arrive at that
$$ R (p, q; r) \leq (R (p - 1, q; r), R (p, q - 1; r), r - 1) + 1 $$
where he states that, as $ R (R (p - 1, q; r), R (p, q - 1; r), r - 1) + 1 $ exists and is finite, therefore $R (p, q; r)$ exists
It is not clear to me, is it possible to use induction hypotheses at the same time?
investigating this induction on two variables, I found that the induction should be done separately.
If we have the statement $P(m,n)$ then we need to prove three things:
Basecase: We need to prove a base case $P(a,b)$ where a is the smallest value for which m is valid and b is the smallest value where n is valid. Induction over m: We assume that $P(k,b)$ is valid for some positive integer $k$. Then we need to prove that $P(k+1,b)$ is valid. Induction over n: We assume that $P(h,k)$ is valid for some positive integers $h,k$. Then we need to prove that $P(h,k+1)$ is valid.
Im sorry for my English,