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We study dense packings of disks and related Gibbs distributions representinghigh-density phases in the hard-core model on unit triangular, honeycomb andsquare lattices. The model is characterized by a Euclidean exclusion distance$D>0$ and a value of fugacity $u>0$. We use the Pirogov-Sinai theory to studythe Gibbs distributions for a general $D$ when $u$ is large: $u>u_0(D)$. Forinfinite sequences of values $D$ we describe a complete high-density phasediagram: it exhibits a multitude of co-existing pure phases, and their numbergrows as $O(D^2)$. For the remaining values of $D$, except for those withsliding, the number of co-existing pure phases is still of the form $E(D)\geqO(D^2)$; however, the exact identification of the pure phases requires anadditional analysis. Such an analysis is performed for a number of typicalexamples, which involves computer-assisted proofs. Consequently, for all values$D>0$ where sliding does not occur, we establish the existence of a phasetransition.  The crucial steps in the study are (i) the identification of periodic groundstates and (ii) the verification of the Peierls bound. This is done by usingconnections with algebraic number theory. In particular, a complete list ofso-called sliding values of $D$ has been specified. As a by-product, we solvethe disk-packing problem on the lattices under consideration. The number andstructure of maximally-dense packings depend on the disk-diameter $D$, unlikethe case of $\mathbb{R}^2$.

The hard-core model on planar lattices: the disk-packing problem and high-density phases