论文标题
避开块的点测序
Block-avoiding point sequencings
论文作者
论文摘要
令$ n $和$ \ ell $为正整数。 Kreher，Stinson和Veitch的最新论文探索了$ n $点上订购三个系统（例如Steiner Triple System，定向三重系统或Mendelsohn Triple System）中点数的问题的变体，因此在$ \ ell $ $ \ ell $ contecectectectecties中不会发生任何块（因此，订购的订单都在订单中（因此是blockelly blockellake blocky blocky blockyavelinging）。我们描述了一种贪婪的算法，该算法表明存在这样的订购，前提是与$ \ ell $相比，$ n $足够大。该算法导致在已知的情况下的点数上提高了点数的界限，但也将结果扩展到更加通用的设置（例如，包括避免设计块的订购）。还建立了这种情况的循环变体的类似结果。 我们构建了Steiner Triple Systems和四倍系统，其中$ \ ell $可能很大，这表明Stinson和Veitch的界限是合理的。此外，我们概括了Stinson-Veitch与更广泛的块设计和循环案例绑定在一起。 克雷尔（Kreher），斯坦森（Stinson）和维奇（Veitch）的结果最初是受艾尔斯帕（Alspach），克雷尔（Kreher）和帕斯汀（Pastine）的结果启发的，克雷尔（Kreher）和帕斯汀（Pastine）（由零和避免阿伯利亚人组的序列驱动）对部分施坦纳三重系统中的点订购的顺序感兴趣，因为没有一个段，没有段是分离片段的段。 alspach〜 \ emph {et al。} \表明，当系统最多包含$ k $成对的分离块时，当点数数量超过$ 15K-5 $时，存在订购。通过使用贪婪的方法，该论文将其改进到$ 9K+O（k^{2/3}）$。
Let $n$ and $\ell$ be positive integers. Recent papers by Kreher, Stinson and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system, directed triple system or Mendelsohn triple system) on $n$ points so that no block occurs in a segment of $\ell$ consecutive entries (thus the ordering is locally block-avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that $n$ is sufficiently large when compared to $\ell$. This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, for example, orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct Steiner triple systems and quadruple systems where $\ell$ can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalise the Stinson--Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson and Veitch were originally inspired by results of Alspach, Kreher and Pastine, who (motivated by zero-sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach~\emph{et al.}\ show that, when the system contains at most $k$ pairwise disjoint blocks, an ordering exists when the number of points is more than $15k-5$. By making use of a greedy approach, the paper improves this bound to $9k+O(k^{2/3})$.
