Hill's Conjecture states that the crossing number $\text{cr}\left(K_n\right)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is $\frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor \frac{n-3}{2}\rfloor =n^4∕64+O\left(n^3\right)$. Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely $n^4∕64+O\left(n^3\right)$, thus matching asymptotically the conjectured value of $\text{cr}\left(K_n\right)$. Let ${\text{cr}}_P\left(G\right)$ denote the crossing number of a graph $G$ in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of $K_n$ is $\left(n^4∕8{\pi }^2\right)+O\left(n^3\right)$. In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if $\underset{n\to \infty }{\lim } {\text{cr}}_P\left(K_n\right)∕n^4=1∕8{\pi }^2$. We construct drawings of $K_n$ in the projective plane that disprove this.

中文翻译：

---

投影平面上完整图形的图形

希尔的猜想指出，交叉数 $\text{铬}（{ķ}_{ñ}）$ 完整图的 ${ķ}_{ñ}$ 在平面上（相当于球体）是 $\frac{\mathrm{1个}}{4}\lfloor \frac{ñ}{\mathrm{2个}}\rfloor \lfloor \frac{ñ-\mathrm{1个}}{\mathrm{2个}}\rfloor \lfloor \frac{ñ-\mathrm{2个}}{\mathrm{2个}}\rfloor \lfloor \frac{ñ-3}{\mathrm{2个}}\rfloor ={ñ}^4∕64+Ø（{ñ}^3）$。Moon证明，在球面图形中，通过随机测地线将点随机分布并连接的预期交叉点数是精确的${ñ}^4∕64+Ø（{ñ}^3）$，从而渐近匹配 $\text{铬}（{ķ}_{ñ}）$。让${\text{铬}}_P（G）$ 表示图的交叉数 $G$在投影平面上。最近，埃尔基斯（Elkies）证明，在自然定义的随机投影平面图中，预期的交叉次数${ķ}_{ñ}$ 是 $（{ñ}^4∕8{\pi }^{\mathrm{2个}}）+Ø（{ñ}^3）$。类似于月亮的结果与希尔的猜想的关系，埃尔基斯问$\underset{ñ\to \infty }{林} {\text{铬}}_P（{ķ}_{ñ}）∕{ñ}^4=\mathrm{1个}∕8{\pi }^{\mathrm{2个}}$。我们绘制的图纸${ķ}_{ñ}$ 在投射平面上证明了这一点。