For every integer n≥ 4 and every rotation system R on an n-element set, at least H(n)=frac14lfloor fracn2rfloor lfloorfracn-12rfloor lfloorfracn-22rfloor lfloorfracn-32rfloor four-element subsets induce a non-planar rotation system; that is, their induced rotation system cannot be realized by a crossing-free drawing of K_4.

Archdeacon’s conjecture

For every integer $n\ge 4$ and every rotation system $R$ on an $n$ -element set, at least $H(n)=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor$ four-element subsets induce a non-planar rotation system; that is, their induced rotation system cannot be realized by a crossing-free drawing of $K_4$ .

References

Primary source

arXiv

Additional references

Some results on Archdeacon's conjecture for rotation systems — arXiv — Arahat Chikkatur, Ji Zeng

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new paper verifies the conjecture in small cases and gets close to its predicted bound asymptotically, but the general conjecture remains open.

Archdeacon’s conjecture proposes a lower bound strengthening Hill’s crossing-number conjecture. The full bound $H(n)$ is not known in general.

September 2026 asymptotic advance

Arahat Chikkatur and Ji Zeng’s preprint reports verification of the bound for $n \le 10$ , asymptotic lower bounds approaching $H(n)$ , and an exact result for a structured class of rotation systems. This is substantial claimed progress, not a complete resolution.

Current status (as of September 2026): The bound is reported for $n \le 10$ and a structured class, with asymptotic lower bounds approaching it, but the full general conjecture remains open.

Sources

No solutions have been posted yet.