Improved Lower Bounds for Van Der Waerden Numbers

Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by $${k^{{b_0}\left( k \right)}}$$ where $${b_0}\left( \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}$$ . We prove a new lower bound of kb(k) with $$b\left( {{c\log \,k}}$$ This done modifying Green’s argument, replacing complicated result about random quadratic forms an elementary probabilistic result.