Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].

中文翻译：

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不可数结构不可分类至双嵌入性

回答 [L. 座右铭 Ros，大型模型的可嵌入性关系的描述性集合理论复杂性，安。纯应用。Logic 164(12) (2013) 1454–1492]，我们证明只要 [Formula: see text] 是满足 [Formula: see text] 的基数，那么 [Formula: see text] 大小的结构之间的可嵌入性关系是强的不变地普遍，因此对于（[公式：见文本]-）分析准阶是完整的。我们还证明，在上述结果中，我们可以进一步将注意力限制在各种自然类别的结构上，包括（广义）树、图或组。这将 [A. Louveau 和 C. Rosendal，完整的解析等价关系，Trans。阿米尔。数学。社会党。357（12）（2005）4839–4866；S.-D。弗里德曼和 L. Motto Ros，分析等价关系和双嵌入性，J. Symbolic Logic 76(1) (2011) 243–266；J. Williams，通用可数 Borel 准序，J. Symbolic Logic 79(3) (2014) 928–954；F. Calderoni 和 L. Motto Ros，组嵌入性的普遍性，Proc。阿米尔。数学。社会党。146 (2018) 1765–1780]。