Title: Alternating Parity Weak Sequencing Show Chinese title

Title: 交替奇偶弱序列

Abstract: A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an ordering $(y_1, \ldots, y_k)$ of its elements such that the partial sums~$s_0, s_1, \ldots, s_k$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i y_j$ for $1 \leq i \leq k$, satisfy $s_i \neq s_j$ whenever and $1 \leq |i-j|\leq t$. In [10] it was proved that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p > 3$ is prime, $e \leq 3$ and $t \leq 6$. Inspired by this result, we show that, if $G$ is the semidirect product of $\mathbb{Z}_p$ and $\mathbb{Z}_2$ and the subset $S$ is balanced, then $S$ admits, regardless of its size, an alternating parity $t$-weak sequencing whenever $p > 3$ is prime and $t \leq 8$. A subset of $G$ is balanced if it contains the same number of even elements and odd elements and an alternating parity ordering alternates even and odd elements. Then using a hybrid approach that combines both Ramsey theory and the probabilistic method we also prove, for groups $G$ that are semidirect products of a generic (non necessarily abelian) group $N$ and $\mathbb{Z}_2$, that all sufficiently large balanced subsets of the non-identity elements admit an alternating parity $t$-weak sequencing. The same procedure works also for studying the weak sequenceability for generic sufficiently large (not necessarily balanced) sets. Here we have been able to prove that, if the size of a subset $S$ of a group $G$ is large enough and if $S$ does not contain $0$, then $S$ is $t$-weakly sequenceable. Show Chinese abstract

Abstract: 群 $(G,+)$ 的子集 $S$ 如果存在一种元素排列方式 $(y_1, \ldots, y_k)$，使得部分和 $s_0, s_1, \ldots, s_k$，由 $s_0 = 0$ 和 $s_i = \sum_{j=1}^i y_j$ 给出且对于 $1 \leq i \leq k$ 成立时满足 $s_i \neq s_j$ 和 $1 \leq |i-j|\leq t$，则称其为 $t$-弱序列可构造的。 在[10]中证明了，如果一个群的阶为$pe$，那么当$p > 3$是素数时，非单位元的所有足够大的子集在$e \leq 3$和$t \leq 6$条件下都是$t$-弱序列可排列的。 受此结果启发，我们证明了，如果 $G$ 是 $\mathbb{Z}_p$ 和 $\mathbb{Z}_2$ 的半直积，并且子集 $S$ 是平衡的，那么只要 $p > 3$ 是素数且 $t \leq 8$，则无论大小如何，$S$ 都存在交替奇偶性的 $t$-弱序列化。 一个集合 $G$ 是平衡的，当且仅当它包含相同数量的偶数元素和奇数元素，并且交替奇偶性排序交替出现偶数和奇数元素。 然后，利用结合了拉姆齐理论和概率方法的混合方法，我们还证明了对于半直积群 $G$，其中 $N$ 是一个通用的（不必然是阿贝尔群）群，$\mathbb{Z}_2$ 是任意的群，所有足够大的非单位元平衡子集都承认交替奇偶性 $t$-弱序列化。 同样的过程也可以用于研究通用的足够大的（不必然是平衡的）集合的弱序列化问题。 在这里我们已经能够证明，如果一个群 $G$ 的子集 $S$ 的大小足够大，并且如果 $S$ 不包含 $0$，那么 $S$ 是 $t$-弱可序列的。