Optimal approximation rate for edge-based Szemerédi regularity

Determine whether the O(1/sqrt(n)) rate for approximating an arbitrary grapheur by a graph with n edges is optimal.

Background

The paper proves an edge-based Szemerédi-type regularity result: every grapheur can be approximated in the W_square metric by a graph sampled or constructed using n edges, with an error of order 1/sqrt(n). The universal constant in the proved bound is explicit.

The unresolved issue is whether the order of this rate can be improved, rather than merely whether the numerical constant can be reduced.

References

Is the rate of $O(1/\sqrt{n})$ for approximating a grapheur by a graph on $n$ edges optimal?

— Graph Limits via Quotients  (2512.23149 - Levin et al., 29 Dec 2025) in Section 6, “Conclusions and Future Directions,” item “Optimal Szemerédi rate”