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Thanks to Roman Kossak and Athar Abdul-Quader whose constructive comments lead to, among other things, a simplification in the proof of Theorem 4.
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Schmerl, J.H. Subsets coded in elementary end extensions. Arch. Math. Logic 53, 571–581 (2014). https://doi.org/10.1007/s00153-014-0381-z
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DOI: https://doi.org/10.1007/s00153-014-0381-z