User blog comment:Hyp cos/Question about weak compactness/@comment-35470197-20180911215601/@comment-35392788-20180913104141

A weakly compact cardinal is a cardinal \(K\) such that for all functions \(f: K^2\mapsto\{0,1\}\), there exists an unbounded subset of \(K\) that consists of either all 0s or all 1s.

From there, I would define a "2-weakly compact" cardinal as such :

A 2-weakly compact cardinal is a cardinal \(K\) such that for all functions \(f: \kappa^2\mapsto\{0,1\}\), where \(\kappa\) is the set of weakly compact cardinals below \(K\), there exists an unbounded subset of \(K\) that consists of either all 0s or all 1s.

With this definition, it is possible that a 2-weakly compact be suited for OCF purposes (as in, completely inaccessible (in the informal sense) using any degree of Mahloness). Or, maybe this is just nonsensical wordsalad, don't quote me on this.