I wish to prove $R(p,p)\leq\frac{2^{2p-2}}{\sqrt{p}}$ combinatorially. I have proved this algebraically through the definition of the binomial coefficient but I would much prefer a proof from combinatorics.
I have managed to get as far as $$R(p,p)\leq\binom{2p-2}{p-1}\leq2^{2p-2}$$ but I am unsure as to why the inequality still holds if we divide by $\sqrt{p}$. Any help would be greatly appreciated!
Thanks.