Extremal problems on the hypercube and the codegree Turán density of complete $r$-graphs

Abstract

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-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 $s_{2m}({\mathbb{Z}}_2^d)\le C_m{2}^{d/m}+O(1)$ when $d\to \infty$, and $s_{2m}({\mathbb{Z}}_2^d)\ge 2^{d/m}+2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Turán density of complete $r$-graphs.

Changes made in response to the referee's suggestions

References in corpus (2)

• On Kemnitz' Conjecture Concerning Lattice Points in the Plane
• Zero-sum Subsequences of Length kq over Finite Abelian p-Groups

Cited by in corpus (2)

• Solving Turán's Tetrahedron Problem for the ${\ell }_2$-Norm
• On generalized Erdős-Ginzburg-Ziv constants for ${\mathbb{Z}}_2^d$