Abstract

Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a well-partial ordering always have a well-ordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a well-partial ordering always has a well-ordered extension.

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