Let $P\subset \Bbb{Z}[X]$ be a finite set of polynomials with constant-term zero. Then, polynomial vdW says:
For eacg finite $r$, there exists some $N=N(P,r)$, such that every $r$-coloring $C:\{1,\dots,N\}\to \{1,\dots,r\}$ of the first $N$ naturals, we can find integers $n,d>0$ such that $C(n+p(d)) = C(n)$ for all $p\in P$ (where the coloring is defined for each of these inputs).
For general sets of polynomials $P$, we have rather poor upper bounds for $N(P,r)$ as $r$ grows (we just know it is primitive-recursive, by Shelah). But for various special cases, like if $P$ is a bunch of monomials of the same degree, then we have double-exponential bounds $N\le \exp(\exp(r^{O_P(1)}))$, which matches our knowledge for the standard van der Waerden problem.
I was wondering, what are some explicit examples of sets $P$ where we don't know good bounds??
Best Answer
Isn't this paper the most up to date? I believe the introduction covers all of the relevant literature.
"Proving a fully general quantitative polynomial Szemerédi theorem remains a very challenging open problem, and effective bounds for sets lacking polynomial progressions of complexity at least one where the polynomials $P_1, \cdots, P_m$ are not homogeneous of the same degree are unknown in the integer setting"