I am a graduate student and we are having a course on discrete mathematics.In the second class,we were taught Ramsey's theorem in its most general form:
Theorem
Given positive integers $k,r,\ell_1,...,\ell_r$,there exists a natural number $n=n(k,r,\ell_1,...,\ell_r)$ such that for any $r$-coloring of $\binom{[n]}{k}$ ,the set of all $k$-subsets of $[n]=\{1,2,...,n\}$ by colours $c_1,c_2,...,c_r$ there exists $i\in \{1,2,...,r\}$ such that there exists an $\ell_i$-subset $S$ all of whose $k$ subsets are of colour $c_i$.[The smallest $n$ satisfying the theorem is denoted by $R(k,r,\ell_1,...,\ell_r)$]
I am not comfortable with the proof of this theorem which uses multiple induction.The proof is as follows:
For $k=1$, we have to look at $1$-subsets of $[n]$ which are $n$ singletons.Take $n=\ell_1+...+\ell_r-r+1$ and by generalized pigeonhole principle we are done. Now assume that $R(k-1,r,\ell_1,...,\ell_r)$ extsts for any $\ell_i$.
Now for any $k$, If $\ell_i=k$ for all $i$,then we are done.Also if some $\ell_i<k$ then we are done.
Now assuming that the numbers $a_i=R(k,r,\ell_1,...,\ell_i-1,...,\ell_r)$ exist we have to show that $R(k,r,\ell_1,...,\ell_r)$ exists.The proof ends by showing that $n=1+R(k,r,a_1,...,a_r)$ works for $k,r,\ell_1,...,\ell_r$.
Now I am not sure how we are able to conclude by showing this.For simple induction,if we know that a statement $P(n)$ is true for $n=1$ and if we know that $P(k)$ is true $\implies P(k+1)$ is also true,then it is certain that $P(2)$ is true as $P(1)$ is true,$P(3)$ is true as $P(2)$ is true and so on,thus for all $n\in \mathbb N$ the statement $P(n)$ is true.
But here I do not follow the logic and I also do not follow which quantity is varying and which is fixed.Can someone please write the induction hypothesis and its conclusion for this multiple induction and explain why it works,like on which set are we doing induction and what is our $P$ statement here and on which variable am I doing induction?