Indestructible Guessing Models and the Continuum

We introduce a stronger version of an ${\omega }_1$-guessing model, which we call an indestructibly ${\omega }_1$-guessing model. The principle IGMP states that there are stationarily many indestructibly ${\omega }_1$-guessing models. This principle, which follows from PFA, captures many of the consequences of PFA, including the Suslin hypothesis and the singular cardinal hypothesis. We prove that IGMP is consistent with the continuum being arbitrarily large.