I was reading about Ramsey numbers lately, and found a lower bound in a book. It stated that:
$ R(k,k) > \binom{k-1}{3} $
However it did not say the proof and I could not come up with a construction. I found that $R(k,k) \geq (k-1)^2 $, but that is not sharp enough, only for large k.
Any ideas how one could prove the firstly mentioned inequality?