{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,21]],"date-time":"2025-12-21T10:03:47Z","timestamp":1766311427780},"reference-count":37,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","funder":[{"name":"Germany's Excellence Strategy EXC 2044","award":["390685587"],"award-info":[{"award-number":["390685587"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2020,4]]},"abstract":"<jats:p> Answering some of the main questions from\u00a0[L. Motto\u00a0Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454\u20131492], 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\u00a0[A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839\u20134866; S.-D. Friedman and L. Motto\u00a0Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243\u2013266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928\u2013954; F. Calderoni and L. Motto\u00a0Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765\u20131780]. <\/jats:p>","DOI":"10.1142\/s0219061320500014","type":"journal-article","created":{"date-parts":[[2019,7,2]],"date-time":"2019-07-02T03:45:59Z","timestamp":1562039159000},"page":"2050001","source":"Crossref","is-referenced-by-count":3,"title":["Uncountable structures are not classifiable up to bi-embeddability"],"prefix":"10.1142","volume":"20","author":[{"given":"Filippo","family":"Calderoni","sequence":"first","affiliation":[{"name":"University of Illinois at Chicago, Department of Mathematics, Statistics, and Computer Science, 851 South Morgan Street, Chicago, IL 60607-7045, US"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heike","family":"Mildenberger","sequence":"additional","affiliation":[{"name":"Abteilung f\u00fcr Mathematische Logik, Mathematisches Institut, Universit\u00e4t Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg im Breisgau, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"Motto Ros","sequence":"additional","affiliation":[{"name":"Dipartimento di matematica \u226aGiuseppe Peano\u226b, Universit\u00e0 di Torino, Via Carlo Alberto 10, 10123 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2019,9,6]]},"reference":[{"key":"S0219061320500014BIB001","author":"Andretta A.","year":"2019","journal-title":"Mem. 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