For any integer $k2,$ the van der Waerden number $W(2,k)$ is the smallest integer $n$ such that any coloring of $\{1,2,$cdots,$n\}$ in two colors gives a monochromatic arithmetic progression with $k$ terms. Prove that $W(2,k)>\frac{2^{k-1}}{e(k+2)}.$ Hint: Prove the contrapositive by using the Lov$$sz Local Lemma.