  We study dense packings of disks and related Gibbs distributions representing
high-density phases in the hard-core model on unit triangular, honeycomb and
square 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 study
the Gibbs distributions for a general $D$ when $u$ is large: $u>u_0(D)$. For
infinite sequences of values $D$ we describe a complete high-density phase
diagram: it exhibits a multitude of co-existing pure phases, and their number
grows as $O(D^2)$. For the remaining values of $D$, except for those with
sliding, the number of co-existing pure phases is still of the form $E(D)\geq
O(D^2)$; however, the exact identification of the pure phases requires an
additional analysis. Such an analysis is performed for a number of typical
examples, which involves computer-assisted proofs. Consequently, for all values
$D>0$ where sliding does not occur, we establish the existence of a phase
transition.
  The crucial steps in the study are (i) the identification of periodic ground
states and (ii) the verification of the Peierls bound. This is done by using
connections with algebraic number theory. In particular, a complete list of
so-called sliding values of $D$ has been specified. As a by-product, we solve
the disk-packing problem on the lattices under consideration. The number and
structure of maximally-dense packings depend on the disk-diameter $D$, unlike
the case of $\mathbb{R}^2$.
