It is known that the Ramsey number $R(4, 4)$ equals 18. Show that $R(4,5) \le 33$.

Question

It is known that the Ramsey number $R(4, 4)$ equals 18. Show that $R(4,5) \le 33$.

I'm stumped by this question, so could someone please offer a hint?

Answer 1

I expect that you know that $R(3,3)=6$ and know or can easily show that $R(2,s)=s$. Use that information and the fact that $R(4,4)=18$ and apply the result that in general $$R(r,x)\le R(r-1,s)+R(r,s-1)\;.$$