On the quasi-ordering of borel linear orders under embeddability

Journal of Symbolic Logic 55 (2):537-560 (1990)
Abstract
We provide partial answers to the following problem: Is the class of Borel linear orders well-quasi-ordered under embeddability? We show that it is indeed the case for those Borel orders which are embeddable in R ω , with the lexicographic ordering. For Borel orders embeddable in R 2 , our proof works in ZFC, but it uses projective determinacy for Borel orders embeddable in some $\mathbf{R}^n, n , and hyperprojective determinacy for the general case
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DOI 10.2307/2274645
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On the Determinacy of Games on Ordinals.Leo A. Harrington & Alexander S. Kechris - 1981 - Annals of Mathematical Logic 20 (2):109-154.

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Borel Quasi-Orderings in Subsystems of Second-Order Arithmetic.Alberto Marcone - 1991 - Annals of Pure and Applied Logic 54 (3):265-291.

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