Inducibility constant of a single vertex in the 2-cube

Determine the inducibility constant \(\lambda(H)\) when \(H=\{(0,0)\}\) is a single vertex of the 2-dimensional hypercube \(Q_2\), improving the best known lower bound \(\lambda(H)\ge 2/3\).

Background

The paper studies inducibility constants $\lambda(H)$, defined as the asymptotically maximal density of copies of a fixed configuration $H\subseteq\{0,1\}^d$ in subsets of high-dimensional hypercubes. Previous work had resolved three of the four non-equivalent configurations in $Q_2$. The remaining configuration is the one consisting of a single vertex, for which only a lower bound was known when this paper was written.