Generalize the density theorem to k-uniform k-partite hypergraphs

Prove that for every integer k≥1 there exists a constant γ(k)>0 such that, for every sufficiently large t, every colouring of a k-uniform k-partite hypergraph of positive density d with vertex parts of size at least (t/d)^{γ(k)t^{k−1}} contains a canonically coloured copy of the complete k-partite k-uniform hypergraph with t vertices in each part.

Background

The paper proves a density-dependent canonical Ramsey theorem for bipartite graphs, corresponding to k=2. The authors propose extending this result to all uniformities as a possible route toward the optimal upper bound for k-partite canonical Ramsey numbers.

An affirmative answer would imply the desired upper bound for all k≥2. The k=1 case follows from the pigeonhole principle, and the k=2 case is established in the paper, so the unresolved content concerns higher uniformities.

References

Is it true that for every integer $k\ge 1$ there exists a constant $\gamma=\gamma(k)>0$ such that for every sufficiently large integer $t$, every colouring of a $k$-uniform $k$-partite hypergraph $H$ of density $d>0$ and vertex parts of size at least $(t/d){\gamma t{k-1}$ admits a canonical copy of $#1{k}{t}$?

— Sharp bounds for off-diagonal and tripartite canonical Ramsey numbers  (2610.03358 - Gvozdić et al., 2 Oct 2026) in Question 5.1, Section 5, “Concluding remarks”