论文标题
关于存在$（n_ {5}）$和$（n_ {6}）$配置的新界限
New bounds on the existence of $(n_{5})$ and $(n_{6})$ configurations: the Grünbaum Calculus revisited
论文作者
论文摘要
“Grünbaum发病率演算”是BrankoGrünbaum引入的操作集合的通用名称，以产生来自各种输入配置的新$（N_ {4}）$配置。在上一篇论文中，我们概括了其中的两个操作，以任意$（n_k）$配置生产操作，并且我们表明，对于每个$ k $，都存在一个整数$ n_ {k {k} $，以至于所有$ n \ geq n_ {k} $均至少有一个$（n_ {k} $ n_ $ n_ $ n_ $ n_ $ n_ $ n__; $ n_ {6} \ leq 7350 $。在本文中，我们进一步扩展了Grünbaum演算。使用这些操作，以及以前已知和新颖的临时构造的集合，我们以$ k = 5 $和$ k = 6 $来完善界限。也就是说，我们表明$ n_5 \ leq 166 $和$ n_ {6} \ leq 585 $。
The "Grünbaum Incidence Calculus" is the common name of a collection of operations introduced by Branko Grünbaum to produce new $(n_{4})$ configurations from various input configurations. In a previous paper, we generalized two of these operations to produce operations on arbitrary $(n_k)$ configurations, and we showed that for each $k$, there exists an integer $N_{k}$ such that for all $n \geq N_{k}$, there exists at least one $(n_{k})$ configuration, with current records $N_{5}\leq 576$ and $N_{6}\leq 7350$. In this paper, we further extend the Grünbaum calculus; using these operations, as well as a collection of previously known and novel ad hoc constructions, we refine the bounds for $k = 5$ and $k = 6$. Namely, we show that $N_5 \leq 166$ and $N_{6}\leq 585$.
