AI 中文总结
本文通过计算验证和理论证明，研究了Archdeacon关于旋转系统非平面四元子集数量的猜想，给出了改进的下界并扩展了其适用范围。
AI 中文摘要
一个在$n$个元素上的旋转系统为每个元素分配其他$n-1$个元素的一个循环顺序。一个四元素子集是非平面的，如果其诱导的旋转系统不能通过$K_4$的无交叉绘图实现。作为关于完全图交叉数的Hill猜想的组合加强，Archdeacon猜想每个在$n$个元素上的旋转系统至少有$H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$个非平面四元素子集。我们通过计算验证了Archdeacon猜想对于$n\leq 10$成立，并表明在这些阶数中每个极值旋转系统都可以通过简单绘图实现。借助计算机辅助，我们证明了每个在$n$个元素上的旋转系统至少有$(8/9 - o(1)) H(n)$个非平面四元素子集。我们还给出了一个手工证明，得到较弱的下界$(2/3-o(1)) H(n)$。最后，扩展了Felsner关于绘图中对径对的最新工作，我们证明了Archdeacon猜想对于对径可壳旋转系统成立。
英文摘要
A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.
