On a property of the $n$-dimensional cube

TL;DR

This paper proves that in any large enough subset of an n-dimensional cube's vertices, either a claw or an 8-cycle can be found, revealing structural properties of high-dimensional cubes.

Contribution

It establishes a new combinatorial property of the n-dimensional cube relating subset size to induced subgraph structures.

Findings

Subsets with at least 2^{n-1}+1 vertices contain a claw or an 8-cycle.

The result holds for all n ≥ 4.

Provides insights into the induced subgraph structures of hypercubes.

Abstract

We show that in any subset of the vertices of $n$-dimensional cube that contains at least $2^{n-1}+1$ vertices ($n\geq 4$), there are four vertices that induce a claw, or there are eight vertices that induce the cycle of length eight.