论文标题
薄弱的Schur分区的建筑
A construction for weak Schur partitions
论文作者
论文摘要
1952年，J.H. Braun声称已经建立了一个公式，为某些整数集的某些分区提供了一个较低的限制，将其分成弱的无额类别。但是，当时没有任何证据或支持构造。在当今的术语中，这一说法等同于为弱的Schur Number $ ws（S）$提供公式化的下限。 $ ws（s）$是最大数字，因此整数从1到$ ws（s）$，$ s $ subset的整数较弱。在一组整数的薄弱的Schur分区中，在任何子集中都不可能有三个不同的成员$ a $ a $ a，$ b $和$ c $，因此$ a+b = c $。本文描述的迭代结构导致类似的公式化下限。尽管与布劳恩（Braun）给出的不同，但它重现了其公式所暗示的结果$ ws（6）\ ge 554 $，并以$ s $的所有较大值超过了它。各种起点可以用作迭代的基础。该结果本身不再引人注目：在其他地方已经证明了$ ws（6）\ ge 642 $。即便如此，人们希望该公式及其基础构造可能对那些对弱的Schur分区和/或与密切相关的线性拉姆西图感兴趣的人可能感兴趣。
In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's terminology, that claim was equivalent to giving a formulaic lower bound for the weak Schur number $WS(s)$. $WS(s)$ is the maximum number such that there exists a weak Schur partition of the integers from 1 to $WS(s)$, into $s$ subsets. In a weak Schur partition of a set of integers, there can be no three distinct members $a$, $b$ and $c$ in any subset, such that $a+b=c$. An iterative construction described in this paper results in a similar formulaic lower bound. Although different from that given by Braun, it reproduces the result $WS(6) \ge 554$ implied by his formula, and exceeds it for all larger values of $s$. Various starting points can be used as a basis for the iterations. This result itself is no longer remarkable: it has been proven elsewhere that $WS(6) \ge 642$. Even so, it is hoped that the formula and its underlying construction may nevertheless be of interest to those interested in weak Schur partitions and/or the closely-related linear Ramsey graphs.
