The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] 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 Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.[Formula: see text] , eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica 26(2) (2006) 183–205], the Ramsey theory of [Formula: see text] had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin. 2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math. 185(1–3) (1998) 137–181]. 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.

中文翻译：

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通用齐次无三角形图的拉姆齐理论

由 Henson 构建的通用齐次无三角形图 [A family of countable homogeneous graphs, Pacific J. Math. 38(1) (1971) 69–83] 并表示为 [公式：见正文]，是 Rado 图的无三角形类似物。虽然 Rado 图的 Ramsey 理论已经完全确立，但从 Erdős-Hajnal-Posá [在无限和有限集中将图强嵌入到彩色图中开始。Vol.[公式：见正文]，编辑。A. Hajnal、R. Rado 和 V. Sós，Colloquia Mathematica Societatis János Bolyai，Vol。10 (North-Holland, 1973), pp. 585–595] 和 Sauer [Coloring subgraphs of the Rado graph, Combinatorica 26(2) (2006) 231–253] 和 Laflamme–Sauer–Vuksanovic [Canonical partitions通用结构，Combinatorica 26(2) (2006) 183–205]，[公式：见文本] 只发展到顶点着色的边界 [P. Komjáth 和 V. Rödl，通用图的着色，图组合。2(1) (1986) 55–60] 和边缘着色 [N. Sauer，可数三角形自由同质图的边分区，离散数学。185（1-3）（1998）137-181]。这是由于缺乏广泛的技术。我们一般解决这个问题：对于每个有限无三角形图 [公式：见文本]，有一个有限数 [公式：见文本]，使得对于 [公式：见文本] 中的 [公式：见文本] 的所有副本的任何着色: see text] 变成有限的多种颜色，有一个 [Formula: see text] 的子图，它又是普遍齐次无三角形，其中着色不超过 [Formula: see text] 颜色。这是省略一些非平凡有限结构的副本的同质结构的第一个这样的结果。