Exponential Lower Bounds on the Generalized Erd\H{o}s-Ginzburg-Ziv Constant

For a finite abelian group $G$, the generalized Erd\H{o}s–Ginzburg–Ziv constant ${\mathsf{s}}_k(G)$ is the smallest $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. If $n=\exp (G)$ is the exponent of $G$, the previously best known bounds for ${\mathsf{s}}_{kn}(C_n^r)$ were linear in $n$ and $r$ when $k\ge 2$. Via a probabilistic argument, we produce the exponential lower bound $${\mathsf{s}}_{2n}(C_n^r)>\frac{n}{2}[1.25-O(n^{-3\mathrm{/}2}){]}^r$$ for $n>0$. For the general case, we show $${\mathsf{s}}_{kn}(C_n^r)>\frac{kn}{4}(1+\frac{1}{ek}+O(\frac{1}{n}){)}^r\mathrm{.}$$