{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T17:45:00Z","timestamp":1775843100382,"version":"3.50.1"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"11","license":[{"start":{"date-parts":[[2014,7,1]],"date-time":"2014-07-01T00:00:00Z","timestamp":1404172800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.<\/p>","DOI":"10.1090\/s0002-9947-2013-05782-0","type":"journal-article","created":{"date-parts":[[2013,7,1]],"date-time":"2013-07-01T08:42:01Z","timestamp":1372668121000},"page":"5783-5804","source":"Crossref","is-referenced-by-count":31,"special_numbering":"942","title":["Additivity of the dp-rank"],"prefix":"10.1090","volume":"365","author":[{"given":"Itay","family":"Kaplan","sequence":"first","affiliation":[]},{"given":"Alf","family":"Onshuus","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Usvyatsov","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,7,1]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"Hans Adler, Strong theories, burden, and weight, preprint (2007).","DOI":"10.1126\/stke.3902007tw202"},{"key":"2","unstructured":"Matthias Aschenbrenner, Alfred Dolich, Dierdre Haskell, Dugald MacPherson, and Sergei Starchenko, Vapnik-chervonenkis density in some theories without the independence property, arXiv:1109.5438 (2011)."},{"key":"3","unstructured":"Vincent Guingona and Cameron Donnay Hill, Local dp-rank and vc-density over indiscernible sequences, arXiv:1108.2554 (2011)."},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1305\/ndjfl\/1040248458","article-title":"Remarks on structure theorems for \ud835\udf14\u2081-saturated models","volume":"36","author":"Hyttinen, Tapani","year":"1995","journal-title":"Notre Dame J. 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