Off-diagonal monotone-path hypergraph Ramsey lower bound

Prove that, for every fixed integer k≥4, the ordered Ramsey number R_<(K^{<(k)}_n,MP^{<(k)}_{k+1}) is at least t_{k-1}(Ω(n)).

Background

The survey relates off-diagonal ordered Ramsey numbers involving complete ordered k-uniform hypergraphs and monotone paths to classical hypergraph Ramsey numbers. It records a strengthening of an Erdős–Hajnal conjecture proposed by Mubayi and Suk.

References

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”