  Since being isolated by Viale and Weiss in 2009, the Guessing Model Property
has emerged as a particularly prominent and powerful consequence of the Proper
Forcing Axiom. In this paper, we investigate connections between variations of
the Guessing Model Property and cardinal arithmetic, broadly construed. We
improve upon results of Viale and Krueger by proving that a weakening of the
Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that,
though the Guessing Model Property is known not to put an upper bound on the
size of the continuum, it does imply that $2^{\omega_1}$ is as small as
possible relative to the value of $2^\omega$. Building on work of Laver, we
prove that, in the extension of any model of $\mathsf{PFA}$ by a measure
algebra, every tree of height and size $\omega_1$ is B-special (a
generalization of specialness introduced by Baumgartner that can also hold of
trees with uncountable branches). Finally, we investigate the impact of forcing
axioms for Suslin and almost Suslin trees on guessing model properties. In
particular, we prove thatif $S$ is a Suslin tree, then the axioms
$\mathsf{PFA}(S)$ and $\mathsf{PFA}(S)[S]$ imply the Guessing Model Property
and the Indestructible Guessing Model Property, respectively, and, if $T^*$ is
an almost Suslin Aronszajn tree, then the axiom $\mathsf{PFA}(T^*)$ implies the
Indestructible Guessing Model Property. This answers a number of questions of
Cox and Krueger.
