论文标题
在安东尼·希尔（Anthony Hill）的猜想中
On a conjecture by Anthony Hill
论文作者
论文摘要
在1950年代，英式画家安东尼·希尔（Anthony Hill）描述了飞机上完整图的图纸$ k_n $，恰好是$ h（n）= \ tfrac {1} {4} {4} \ lfloor \ tfrac {n} {n} {n} {2} {2} \ rfloor \ rfloor \ \ lfloor \ \ lfloor \，\ lfloor \ tfrac} \ lfloor \ tfrac {n-2} {2} \ rfloor \，\ lfloor \ tfrac {n-3} {2} {2} \ rfloor $ $$交叉。猜想是这个数字最小的可能性，尽管做出了严重的努力，但猜想仍然广泛开放。 Blažek和Koman在1963年发现了另一种具有相同数量的交叉点的$ k_n $的方法。在本说明中，我们首次提供了非常普遍的图纸结构，这些图纸达到了相同的绑定。令人惊讶的是，证明非常短，也可能有资格作为“书籍证明”。特别是，它对1968年Moon发现的现象提供了一个非常简单的解释，即在$ \ rr^3 $ in $ \ rr^3 $中随机的$ n $点$ n $点$ \ ss^2 $ in GeoDesics In the Geodesics加入，从而产生了一张图纸，其杂交数量不可分割地接近山丘值$ h（n）$。
In the 1950's, English painter Anthony Hill described drawings of complete graphs $K_n$ in the plane having precisely $$H(n) = \tfrac{1}{4}\lfloor \tfrac{n}{2}\rfloor \, \lfloor \tfrac{n-1}{2}\rfloor \, \lfloor \tfrac{n-2}{2}\rfloor \,\lfloor \tfrac{n-3}{2}\rfloor$$ crossings. It became a conjecture that this number is minimum possible and, despite serious efforts, the conjecture is still widely open. Another way of drawing $K_n$ with the same number of crossings was found by Blažek and Koman in 1963. In this note we provide, for the first time, a very general construction of drawings attaining the same bound. Surprisingly, the proof is extremely short and may as well qualify as a "book proof". In particular, it gives a very simple explanation of the phenomenon discovered by Moon in 1968 that a random set of $n$ points on the unit sphere $\SS^2$ in $\RR^3$ joined by geodesics gives rise to a drawing whose number of crossings asymptotically approaches the Hill value $H(n)$.
