Extension of the midrange crossing constant theorem

Determine the weakest conditions on the number of edges e in terms of the number of vertices n under which the limit defining the midrange crossing constant holds, beyond the condition n << e << n^2.

Background

For graphs with n vertices and e edges in the midrange regime n << e << n2, the minimum crossing number satisfies a normalized limit of the form κ(n,e)n2/e3 → c > 0. The constant c is the midrange crossing constant, currently bounded between approximately 0.036 and 0.09.

The paper asks whether the asymptotic hypothesis can be weakened to fixed linear and quadratic density bounds, C1n ≤ e ≤ C2n2, and more generally seeks the weakest edge-density assumptions under which the same limiting statement remains valid. Necessary restrictions mentioned in the paper are e > (4+ε)n and e < (1/2−ε)n2.

References

It is not known whether the condition $n\ll e\ll n2$ can be replaced by a weaker condition, say, $C_1n\le e\le C_2n2$. It is not hard to see that $(4+\varepsilon)n<e$ and $(1/2-\varepsilon)n^2>e$ are necessary conditions. Find the weakest conditions on $e$ in terms of $n$ such that the statement of Theorem \ref{midrange} holds.

— Generalizations of the Crossing Lemma  (2509.14074 - Toth, 17 Sep 2025) in Section 5, Open problems, item 1; see also Section 2, following Theorem 2.1