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17. Small hexagons in simple topological graphs

    1. Small hexagons in simple topological graphs

          Let $h(n)$ be the smallest area enclosed by a plane $6$-cycle inside any simple topological graph on $n$ vertices drawn inside the unit square. It is known that $ \Omega(\frac{1}{n}) \leq h(n) \leq (\log n)^{- \frac{1}{4} + o(1)} $.

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      Problem 17.1.

      [Ji Zeng] Prove that $h(n) \leq O(\frac{1}{n})$.
          Note: 1. The desired statement holds for $4$-cycles. 2. There exist drawings where all odd (plane) cycles enclose regions with area arbitrarily close to $1$. 3. There exist drawings where all (plane) cycles enclose regions with area at least $\Omega(1/n)$.

      Reference: J. Zeng. Note on disjoint faces in simple topological graphs. Preprint. arXiv:2308.04742.

          Cite this as: AimPL: Albertson conjecture and related problems, available at http://aimpl.org/albertson.