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We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudl\u00e1k\u2013R\u00f6dl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin\u2013Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061316500033","type":"journal-article","created":{"date-parts":[[2016,6,10]],"date-time":"2016-06-10T04:25:40Z","timestamp":1465532740000},"page":"1650003","source":"Crossref","is-referenced-by-count":9,"title":["Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems"],"prefix":"10.1142","volume":"16","author":[{"given":"Natasha","family":"Dobrinen","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Denver, 2280 S Vine St, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2016,7,4]]},"reference":[{"key":"S0219061316500033BIB002","doi-asserted-by":"crossref","DOI":"10.1007\/3-7643-7360-1","volume-title":"Ramsey Methods in Analysis","author":"Argyros S. 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