Extremal problems on the hypercube and the codegree Tur\'an density of complete $r$-graphs

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erd\H{o}s-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We show that $s_{2m}({\mathbb{Z}}_2^d)\le C_m{2}^{d\mathrm{/}m}+O(1)$ when $d\to \mathrm{\infty }$, and $s_{2m}({\mathbb{Z}}_2^d)\ge 2^{d\mathrm{/}m}+2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Tur'{a}n density of complete $r$-graphs.