The well-known hard-core model on the triangular lattice $\mathbb A$ is an interaction model defined on the independent sets of the lattice, parameterized by an activity parameter $λ> 0$. In this work, we consider an extension of this model to maximal independent sets, which we call the maximal hard-core model. We show that in the high-activity regime ($λ> 10^6$), the model admits at least two Gibbs measures on maximal independent sets in $\mathbb A$. In the low-activity regime ($λ$ close to $0$), we apply Pirogov-Sinai theory to characterize extremal periodic Gibbs measures of the model. Finally, we derive bounds on the capacity of a recoverable system on the lattice associated with the maximal hard-core model.

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