Off-diagonal hypergraph ordered Ramsey conjecture

Prove that for every fixed integer $k\ge4$, the off-diagonal ordered Ramsey number $R_<(K^{<(k)}_n,MP^{<(k)}_{k+1})$ is at least $t_{k-1}(\Omega(n))$.

Background

The conjecture strengthens a classical conjecture concerning off-diagonal Ramsey numbers of complete uniform hypergraphs. It predicts a tower-type lower bound for an ordered complete hypergraph versus a monotone tight path.

References

Mubayi and Suk also introduced the following strengthening of a conjecture of Erd\H{o}s and Hajnal about off-diagonal Ramsey numbers of complete $k$-uniform hypergraphs.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture in Subsection “Connections to Hypergraph Ramsey Numbers”

Mubayi and Suk also introduced the following strengthening of a conjecture of Erd\H{o}s and Hajnal about off-diagonal Ramsey numbers of complete $k$-uniform hypergraphs.

For any fixed integer $k \geq 4$, R_<(K{<(k)},MP{<(k)}_{k+1}) \geq t_{k-1}(\Omega(n)).

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 3, subsection “Connections to Hypergraph Ramsey Numbers,” Conjecture cited as [mubSuk17]