The Van der Waerden number $W(r,k)$ is the smallest positive integer $n$ such that every $r$-color of $\{1,2,3,\dots,n\}$ contains a monochromatic arithmetic progression of length $k$.
I am looking for the best known lower bound for $W(r,k)$. In the case $r=2$ the best bound is due to Berlekamp: for $p$ a prime, $p \cdot 2^p \leq W(2,p+1)$, which can be extended to a general bound for this $r=2$ case.
I can not find a reference for general $r$, though. I have checked recent surveys, Wikipedia, Rudiments of Ramsey theory (by Graham and Butler), Ramsey Theory (by Graham, Rothschild and Spencer), and have found no reference to a general lower bound.
Has no one found any reasonable lower bound? Or am I skipping over it somehow?
Any help or reference would be greatly appreciated.
Landman and Robertson's Ramsey Theory on the Integers cites Everts's 1977 Ph.D. thesis Colorings of sets (which, as usual, is hard to find online) for the bound
$$W(r,k) > \frac{k r^k}{e(k+1)^2}$$
and also gives the mostly-tighter result, due to Shabanov (which might be this article, though they cite a different paper from 2010), that $$W(r,k) > \left(\frac{\sqrt{6}-2}{4}\right) \frac{r^{k-1}}{\sqrt{k\ln k}}\left(1 - \frac1k\right).$$ The above, they say, is the best known bound for general $r$ and $k$, but for primes $p$ and $q$, the bound $W(q,p+1) \ge p(q^p-1)+1$ "can be easily derived from a similar proof" in Graham, Rothschild, and Spencer's Ramsey Theory.
It's worth comparing all this to the naive probabilistic lower bound: the set $\{1,2,\dots,n\}$ contains fewer than $\binom{n}{2}$ progressions of length $k$ (because $\binom{n}{2}$ is the number of ways to choose a start and end with no regard to whether that's valid), each of which is monochromatic with probability $r^{-(k-1)}$. So if $n < r^{\frac{k-1}{2}}$, then the expected number of monochromatic arithmetic progressions in a random coloring is less than $1$, and with positive probability a random coloring avoids them all.