Diamond on trees, part 1

We explore a generalization of the diamond principle to nonspecial trees of height omega-one, based on the Todorcevic–Brodsky notion of stationarity. We propose this generalized principle as a structural invariant that naturally partitions nonspecial omega-one-trees, according to their “thickness,” into three categories: trees where the principle holds outright, provably in ZFC; “thin” trees where it is equivalent to the classical diamond principle; and certain nonspecial Aronszajn trees where it is strictly stronger than diamond. The talk will also cover the consistency results for this last category, introducing two new forcing properties — closure relative to a set S, and strategic closure in models — both of which are preserved under countable support iterations.