Characterizing large cardinals in terms of layered posets

Given an uncountable regular cardinal $κ$, a partial order is $κ$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $κ$ is stationary in $\mathcal{P}_κ(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $κ$ is weakly compact if and only if every partial order satisfying the $κ$-chain condition is $κ$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $κ$-Knaster partial orders are $κ$-stationarily layered implies that $κ$ is a Mahlo cardinal and every stationary subset of $κ$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

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