Determine the maximum number of edges with exactly one crossing

Determine the exact value of the maximum number of edges with exactly one crossing in a rectilinear drawing of the complete graph K_n for every n; alternatively, determine whether the limit of the normalized quantity overline{max} e_1(K_n)/n exists and, if so, compute its value.

Background

The paper proves lower and upper bounds for the maximum number of edges involved in exactly one crossing in a rectilinear drawing of K_n. Specifically, it establishes a lower bound of approximately 3n/2 and derives an upper bound from prior work on the maximum number of edges involved in at most one crossing. The exact value therefore remains unresolved, as does the asymptotic constant if the normalized maximum has a limit.

References

We leave the issue of closing the gap between the upper and the lower bounds as an open problem. Determine $\overline\max\ {e_1(K_n)}$ for all $n$. Less ambitiously, does the limit $\lim_{n\rightarrow \infty}\overline\max\ \frac{e_1(K_n)}{n}$ exist? If so, what is its value?

— On the crossing profile of rectilinear drawings of $K_n$  (2501.04980 - Chen et al., 9 Jan 2025) in Problem following the discussion of overline{max} e_1(K_n), Section 3.3, "About overline{max} e_1(K_n)"