  The classical hard core model from statistical physics, with activity
$\lambda > 0$ and capacity $C=1$, on a graph $G$, concerns a probability
measure on the set ${\mathcal I}(G)$ of independent sets of $G$, with the
measure of each independent set $I \in {\mathcal I}(G)$ being proportional to
$\lambda^{|I|}$.
  Ramanan et al. proposed a generalization of the hard core model as an
idealized model of multicasting in communication networks. In this
generalization, the {\em multi-state} hard core model, the capacity $C$ is
allowed to be a positive integer, and a configuration in the model is an
assignment of states from $\{0,\ldots,C\}$ to $V(G)$ (the set of nodes of $G$)
subject to the constraint that the states of adjacent nodes may not sum to more
than $C$. The activity associated to state $i$ is $\lambda^{i}$, so that the
probability of a configuration $\sigma:V(G)\rightarrow \{0,\ldots, C\}$ is
proportional to $\lambda^{\sum_{v \in V(G)} \sigma(v)}$.
  In this work, we consider this generalization when $G$ is an infinite rooted
$b$-ary tree and prove rigorously some of the conjectures made by Ramanan et
al. In particular, we show that the $C=2$ model exhibits a (first-order) phase
transition at a larger value of $\lambda$ than the $C=1$ model exhibits its
(second-order) phase transition. In addition, for large $b$ we identify a short
interval of values for $\lambda$ above which the model exhibits phase
co-existence and below which there is phase uniqueness. For odd $C$, this
transition occurs in the region of $\lambda = (e/b)^{1/\ceil{C/2}}$, while for
even $C$, it occurs around $\lambda=(\log b/b(C+2))^{2/(C+2)}$. In the latter
case, the transition is first-order.
