Alternating Parity Weak Sequencing

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.