On a property of the $n$-dimensional cube

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.

💡 Research Summary

The paper investigates an extremal property of the $n$‑dimensional hypercube $Q_n$. It proves that any vertex set $V’\subseteq V(Q_n)$ with at least $2^{,n-1}+1$ elements (for $n\ge4$) must contain either an induced claw $K_{1,3}$ or an induced simple cycle of length eight. The result is a Ramsey‑type statement for hypercubes: a relatively modest density of vertices already forces one of two small substructures.

Main theorem.
For $n\ge4$, if $|V’|\ge2^{,n-1}+1$, then either (a) four vertices of $V’$ induce a $K_{1,3}$ (a claw) or (b) eight vertices of $V’$ induce a $C_8$ (an 8‑cycle).

Proof strategy.
The authors use induction on $n$. The base case $n=4$ is handled by an exhaustive case analysis. $Q_4$ is split into two opposite $Q_3$ subcubes \