A Generalization of the Methods of Brass, Harboth, and Nieborg

TL;DR

This paper generalizes previous methods to construct large $Q_3$-free subgraphs within hypercubes, extending the understanding of extremal properties of such graphs.

Contribution

It introduces new constructions of $Q_3$-free subgraphs of hypercubes for small dimensions, building on earlier work that focused on $Q_4$-free subgraphs.

Findings

Constructed large $Q_3$-free subgraphs for small $n$

Extended methods from previous $Q_4$-free results

Provided insights into hypercube subgraph extremal properties

Abstract

In 1995, Brass, Harborth and Nienborg disproved a conjecture of Erd\H{o}s when they showed that a $C_4$-free subgraph of the hypercube, $Q_n$, can have at least $(\frac 12 +\omega(1))e(Q_n)$ edges. In this paper, we generalize the idea of Brass, Harborth and Nienborg to provide good constructions of $Q_3$-free subgraphs of $Q_n$ for some small values of $n$.