It seems to me that the axiom is equivalent to GCH:
Assume GCH. Let $\lambda<\kappa$ be infinite cardinals. Let $L$ be a total order of cardinality $\kappa$. We want to prove that $L$ contains a copy of $\lambda$ or of $\lambda$ reversed.
Case 1) $\kappa$ is limit. From GCH, $\kappa$ is strong limit and the result follows from Erdos-Rado.
Case 2) $\kappa$ is a successor of a successor: $\kappa=\lambda^{++}$. Then, from GCH and Erdos-Rado, $L$ contains a copy of $\lambda^+$ or of $\lambda^+$ reversed, and this is enough.
Case 3) $\kappa$ is a successor of a limit $\lambda$. From GCH and the version of Erdos-Rado given in Levy, Basic set theory, theorem 3.13, chapter IX, $\lambda^+\rightarrow (\lambda)^2_2$. Since $\kappa=\lambda^+$, we have that $L$ contains a copy of $\lambda$ or of $\lambda$ reversed, and this is enough.