I am studying Ramsey Theorem from Introduction to Graph Theory by Douglas West, 2nd edition. I could not get some part of its proof. Here is the theorem,
8.3.7. Theorem. (Ramsey [1930]) Given positive integers $r$ and $p_1, ..., p_k$, there exists an integer $N$ such that every $k$-coloring of $\binom{[N]}{r}$ yields an $i$-homogeneous set of size $p_i$ for some $i$.
Double induction is used in the proof, on $r$ and $\sum p_i$. I could not understand the part below in the proof.
Basis step: Some quota $p_i$ is less than $r$. In this case, a set of $p_i$ objects contains no $r$-sets, so vacuously its $r$-sets all have color $i$. Hence $R(p_1, ..., p_k; r) = min\{p_1, . . . , p_k\}$ when $min\{p_1, ... , p_k\} < r$.
How $p_i$ being less than $r$ causes this situation ? I didn't understand why $r$ and $p_i$ are related like above. We're choosing $r$-element subsets of $[N]$, so there will be $\frac{N!}{r!(N-r)!}$ subsets to be partitioned into $k$ classes.
Suppose that there is an $i$ such that $p_i<r$, and let $c$ be a $k$-coloring of $\binom{[N]}r$. Let $S\subseteq[N]$ be any set of $p_i$ objects. Then $S$ has no $r$-subsets, so it is vacuously true that every $r$-subset of $S$ has color $i$. If that point is where you’re having trouble, ask yourself what would be needed to show that $S$ was not $i$-homogeneous: you’d have to fine two $r$-subsets of $S$ that were colored differently. But you can’t find even one $r$-subset of $S$, so you certainly can’t find two with different colors!
Thus, in this case $R(p_1,\ldots,p_k;r)\le p_i$. I write ‘$\le$’ because there might be an even smaller $p_j$, in which case the same argument would apply to show that $R(p_1,\ldots,p_k;r)\le p_j$. Thus, we might as well simply pick the smallest quota and observe that
$$R(p_1,\ldots,p_k;r)=\min\{p_1,\ldots,p_k\}$$
if $\min\{p_1,\ldots,p_k\}<r$.