Avoiding zero-sum subsequences of prescribed length over the integers

Let $t$ and $k$ be a positive integers, and let $I_k={i\in $\mathbb{Z}$:; -k\leq i\leq k}$\mathrm{.}Let$$\mathsf{s}$'_t(I_k)$bethesmallestpositiveinteger$\ell$suchthateveryzero-sumsequence$S$over$I_k$oflength$|S|\ge \ell$containsazero-sumsubsequenceoflength$t$\mathrm{.}Ifnosuch$\ell$ exists, then let ${\mathsf{s}}_t^{\mathrm{\prime }}(I_k)=\mathrm{\infty }$. In this paper, we prove that ${\mathsf{s}}_t^{\mathrm{\prime }}(I_k)$ is finite if and only if every integer in $[1,D(I_k)]$ divides $t$, where $D(I_k)=\max \{2,2k-1\}$ is the Davenport constant of $I_k$. Moreover, we prove that if ${\mathsf{s}}_t^{\mathrm{\prime }}(I_k)$ is finite, then $t+k(k-1)\le {\mathsf{s}}_t^{\mathrm{\prime }}(I_k)\le t+(2k-2)(2k-3)$. We also show that ${\mathsf{s}}_t^{\mathrm{\prime }}(I_k)=t+k(k-1)$ holds for $k\le 3$ and conjecture that this equality holds for any $k\ge 1$.