{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T08:17:15Z","timestamp":1782980235153,"version":"3.54.5"},"reference-count":36,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","funder":[{"name":"National Science Foundation","award":["DMS-1301665"],"award-info":[{"award-number":["DMS-1301665"]}]},{"name":"National Science Foundation","award":["DMS-1600781"],"award-info":[{"award-number":["DMS-1600781"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2020,8]]},"abstract":"<jats:p> The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69\u201383] and denoted [Formula: see text], is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erd\u0151s\u2013Hajnal\u2013Pos\u00e1 [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.[Formula: see text] , eds. A. Hajnal, R. Rado and V. S\u00f3s, Colloquia Mathematica Societatis J\u00e1nos Bolyai, Vol. 10 (North-Holland, 1973), pp. 585\u2013595] and culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006) 231\u2013253] and Laflamme\u2013Sauer\u2013Vuksanovic [Canonical partitions of universal structures, Combinatorica 26(2) (2006) 183\u2013205], the Ramsey theory of [Formula: see text] had only progressed to bounds for vertex colorings [P. Komj\u00e1th and V. R\u00f6dl, Coloring of universal graphs, Graphs Combin. 2(1) (1986) 55\u201360] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math. 185(1\u20133) (1998) 137\u2013181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph [Formula: see text], there is a finite number [Formula: see text] such that for any coloring of all copies of [Formula: see text] in [Formula: see text] into finitely many colors, there is a subgraph of [Formula: see text] which is again universal homogeneous triangle-free in which the coloring takes no more than [Formula: see text] colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code [Formula: see text] and the development of their Ramsey theory. <\/jats:p>","DOI":"10.1142\/s0219061320500129","type":"journal-article","created":{"date-parts":[[2019,12,6]],"date-time":"2019-12-06T06:09:36Z","timestamp":1575612576000},"page":"2050012","source":"Crossref","is-referenced-by-count":17,"title":["The Ramsey theory of the universal homogeneous triangle-free graph"],"prefix":"10.1142","volume":"20","author":[{"given":"Natasha","family":"Dobrinen","sequence":"first","affiliation":[{"name":"University of Denver, Department of Mathematics, 2390 S. 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