For a graph G, the density of G, denoted D(G), is the maximum ratio of the number of edges to the number of vertices ranging over all subgraphs of G. For a class \(\mathcal {F}\) of graphs, the value \(D(\mathcal {F})\) is the supremum of densities of graphs in \(\mathcal {F}\). A k-edge-colored graph is a finite, simple graph with edges labeled by numbers \(1,\ldots ,k\). A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class \(\mathcal {F}\) of graphs, a k-edge-colored graph \(\mathbb {H}\) (not necessarily with the underlying graph in \(\mathcal {F}\)) is k-universal for \(\mathcal {F}\) when any k-edge-colored graph with the underlying graph in \(\mathcal {F}\) admits a homomorphism to \(\mathbb {H}\). Such graphs are known to exist exactly for classes \(\mathcal {F}\) of graphs with acyclic chromatic number bounded by a constant. The minimum number of vertices in a k-uniform graph for a class \(\mathcal {F}\) is known to be \(\Omega (k^{D(\mathcal {F})})\) and \(O(k^{{\left\lceil D(\mathcal {F}) \right\rceil }})\). In this paper we close the gap by improving the upper bound to \(O(k^{D(\mathcal {F})})\) for any rational \(D(\mathcal {F})\).

对于图G，表示为D（G）的G密度是边的数量与在G的所有子图中分布的顶点数量的最大比率。对于一类\（\ mathcal {F} \），值\（D（\ mathcal {F}）\）是\（\ mathcal {F} \）中图的密度的总和 。甲ķ _edge时色图表是一个有限的，简单的图形与由数字标记的边缘\（1，\ ldots中，k \） 。顶点集为1 k的函数如果将任何边缘的端点映射到由相同颜色的边缘连接的两个不同顶点，则-edge-colored图到另一图是同态的。给定一类\（\ mathcal {F} \）图，一个 k边色图\（\ mathbb {H} \）（不一定带有\（\ mathcal {F} \）中的基础图 ）ķ -通用为\（\ mathcal {F} \）当任何ķ与底层的图_edge时色图表\（\ mathcal {F} \）承认同态到\（\ mathbb {H} \） 。已知这类图确实存在于类\（\ mathcal {F} \）中具有以常数为界的非循环色数的图的集合。类\（\ mathcal {F} \）在k均匀图中的最小顶点数已知为\（\ Omega（k ^ {D（\ mathcal {F}）}））\）和\（ O（k ^ {{\ left \ lceil D（\ mathcal {F}）\ right \ rceil}}）\）。在本文中，我们通过提高任何有理\（D（\ mathcal {F}）\）的\（O（k ^ {D（\ mathcal {F}）}）\）的上限来缩小差距。