The block-permutation conjecture

For positive integers $m$ and $n$, let $[m]=\{1,2,\ldots,m\}$ and let $[m]^n$ denote the words of length $n$ over $[m]$. A block permutation set in $[m]^n$ is obtained by choosing pairwise disjoint blocks $I_1,\ldots,I_m\subset[n]$, fixing all coordinates outside their union, and taking all words obtained by assigning a permutation of $1,\ldots,m$ to the blocks. Its degree is $\sum_{j=1}^m|I_j|$.

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

This is the fixed-block-size Hales--Jewett-type formulation arising from the group-action conjecture. It asks for a monochromatic family containing all permutations of the alphabet across prescribed blocks; the paper presents it as a conjectural combinatorial reformulation.