Improve bounds for off-diagonal canonical Ramsey numbers

Determine whether the methods for off-diagonal Erdős–Rado numbers can improve either of the general bounds in Proposition 2.1, or otherwise provide a sharper understanding of off-diagonal canonical Ramsey numbers, including their hypergraph analogues.

Background

The paper establishes that the two general bounds for ER(a,b,c), with a,c≥3 and b≥4, are sharp up to constants in the exponent in different parameter regimes. It does not resolve whether the general bounds can be improved outside those regimes or whether a more complete description is possible.

The authors explicitly raise the possibility of extending these questions to hypergraphs, while noting that they are unaware of progress in that direction.

References

We wonder if our methods can be extended to improve either of the bounds of (\ref{eq-lef-rodl-bounds}), or at least give a better understanding of the problem. In this regard, the study of off-diagonal Erdős-Rado numbers in hypergraphs might as well turn out to be of some interest. We are not aware, however, of any progress in this direction.

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