On a relation between λ-full well-ordered sets and weakly compact cardinals

We prove, via transfinite recursion, the existence, inside any linearly ordered set of appropriate regular cardinality λ, of a particular kind of well-ordered subsets characterized by the property of λ-fullness. Let H be a set of regular cardinals: by using our results about well-ordered λ-full sets we show that if inf H is a weakly compact cardinal, then, for every LOTS X, H-compactness is equivalent to the nonexistence of gaps of types in H.