**Let $q_1$, $q_2$, ..., $q_k$, t be positive integers, where $q_1$≥t, $q_2$≥t, ..., $q_k$≥t. Let m be the largest of $q_1$, $q_2$, ..., $q_k$. Show that**

**$r_t$(m, m, ..., m) ≥ $r_t$($q_1$, $q_2$, ..., $q_k$).**

**Conclude that to prove Ramsey's theorem, it is enough to prove it in the case that $q_1$ = $q_2$ = ... = $q_k$.**

This problem is from Brualdi's *Introductory Combinatorics* p.85.

I have a strong intution to apply induction on t, and I got a good grasp on the case t=1 (in which case the problem is reduced to pigeonhole principle).

But, I'm stuck with the inference from t=n-1 to t=n. How to proceed here?

Thanks.

Notation : By $r_t$(q), Brualdi means the Ramsey number of $K^t_q$, denoting the collection of all subsets of t elements of a set of q elements. (Thus, by this notation, $K^2_3$ denotes the complete graph of triangle.)