Compactness principles, GCH and anti-guessing principles

Compactness principles often times imply guessing principles, for example, if kappa is measurable, then the diamond principle holds at kappa. Though small uncountable cardinals (like omega_2) is not a large cardinal, there are many compactness principles that can consistently hold at omega_2. A theorem of Shelah states in the model where GCH holds and omega_2 is ``sufficiently’’ compact, then the diamond principle holds at the points in omega_2 having cofinality omega_1. We will demonstrate a scenario where omega_2 is still pretty ``compact’’, but the diamond principle as above fails in a rather severe way. This shows Shelah’s theorem is optimal in some sense. Some relationships between our model and the model of GCH + aleph_2-Souslin hypothesis (if it exists) will also be discussed.