The block sets conjecture

Let $[m]=\{1,\ldots,m\}$, and let a template over $[m]$ be a non-decreasing word $T\in[m]^l$ for some positive integer $l$. A block set with template $T$ in $[m]^n$ is formed by choosing pairwise disjoint blocks $I_1,\ldots,I_l\subset[n]$ of a common size $d$, fixing the remaining coordinates, and allowing the coordinates on the blocks to realize every permutation of the letters of $T$.

Block sets conjecture. For every pair of positive integers $m$ and $k$ and every template $T$ over $[m]$, there exist positive integers $n$ and $d$ such that every $k$-colouring of $[m]^n$ contains a monochromatic block set of degree $d$ with template $T$.

The conjecture is a purely combinatorial statement that implies every subtransitive set is Ramsey. It is also described as an abstract version of the conjecture that all transitive sets are Ramsey, and remains open.