Determine whether there exists an absolute constant c>0 such that, for every integer d≥ 1, there is a constant C_d>0 for which every weakly d-degenerate ordered 3-uniform hypergraph H on t vertices satisfies r_<bigl(H,K_3^(3)(n)bigr)≤ t,2^C_d n^2-c/d for every positive integer n.

Balko–Vizer ordered Ramsey problem

The problem asks for sharp ordered Ramsey bounds for ordered $3$ -uniform hypergraphs with bounded degree, strengthening the earlier Balko–Vizer estimates. The central question was whether the gap between the known upper and lower bounds could be closed.

Known results

Balko and Vizer, 2021: for bounded maximum degree and interval chromatic number $3$ , they proved a subquadratic exponential upper bound, while a substantial gap with lower bounds remained.

September 2026 claimed resolution

Wen Chen, Zihan He, Qizhong Lin, and Meng Liu claim an exponential bound with exponent $n^{2-c/d}$ in the weakly degenerate ordered $3$ -uniform setting, and show that bounded weak degeneracy cannot generally be replaced by bounded standard degeneracy. This appears to answer the cited problem, but the claim is unverified.

Current status (as of September 2026): The original gap is claimed to be closed for the stated weakly degenerate ordered $3$ -uniform setting; independent verification is pending, and the result does not settle the broader ordered-hypergraph question.

Balko–Vizer ordered Ramsey problem

Primary source

Additional references

Progress summary

Solutions 0