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

TL;DR

This paper investigates extremal problems on the hypercube, establishing bounds on the codegree Turán density of complete r-graphs by leveraging properties of the generalized Erdős-Ginzburg-Ziv constant in finite abelian groups.

Contribution

It provides new bounds on the codegree Turán density of complete r-graphs using results related to the generalized Erdős-Ginzburg-Ziv constant for groups.

Findings

Bound s_{2m}(Z_2^d) by C_m 2^{d/m} + O(1) as d→∞

Lower bound s_{2m}(Z_2^d) ≥ 2^{d/m} + 2m-1 when d=km

Derived new bounds for 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 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 $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$ when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$ 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.