Gowers's polynomial density Hales–Jewett conjecture

Let $k,d$ be positive integers, let $\delta>0$, and let $n$ be a positive integer. Write $[n]=\{1,\ldots,n\}$ and

$$\mathcal{K}=[k]^{[n]}\times\dots\times[k]^{[n]^d}.$$

For a subset of a finite set, its density is its cardinality divided by the cardinality of the ambient set. Gowers's polynomial density Hales–Jewett conjecture. If $n$ is large enough depending on $k,d,\delta$ only, then every subset $\mathcal{A}\subseteq\mathcal{K}$ with density at least $\delta$ contains a non-empty subset $S\subset[n]$ and an element

$$y\in[k]^{[n]\setminus S}\cup\dots\cup[k]^{[n]^d\setminus S^d}$$

such that whenever the coordinates of $x\in\mathcal{K}$ are constant on each of the sets $S,\dots,S^d$ and coincide with those of $y$ outside these sets, one has $x\in\mathcal{A}$. This is a proposed polynomial analogue of the density Hales–Jewett theorem; the paper notes that the case $k=2$ is an important open step, while the formulation also connects to arithmetic-progression phenomena for $k\ge3$.