Closing in on Hill's Conjecture

@article{Balogh2019ClosingIO,
  title={Closing in on Hill's Conjecture},
  author={J. Balogh and B. Lidick{\'y} and G. Salazar},
  journal={SIAM J. Discret. Math.},
  year={2019},
  volume={33},
  pages={1261-1276}
}
Borrowing L\'aszl\'o Sz\'ekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) := \frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$, for all $n\ge 3$. This has been verified only for $n\le 12$. Using flag algebras, Norin and Zwols obtained the best known asymptotic… Expand
Maximum Number of Almost Similar Triangles in the Plane
A survey of graphs with known or bounded crossing numbers

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