论文标题
无限图的拉姆西上部密度
Ramsey upper density of infinite graphs
论文作者
论文摘要
对于固定的无限图$ h $，我们研究了单色子图的最大密度至$ h $，这些密度可以在每两彩的边缘$ k _ {\ mathbb {n}} $中找到。这称为Ramsey上部密度为$ h $，由Erdős和Galvin引入。最近，确定了无限路径的Ramsey上部密度。在这里，我们发现了所有本地有限图$ h $最高2倍的密度的价值，回答了Debiasio和McKenney的问题。 我们还发现了包括所有本地森林在内的宽两部分图的确切密度。我们的方法将此问题与解决连续功能的优化问题有关。我们表明，在某些条件下，密度仅取决于$ h $的色数，$ h $的组件数量以及扩展比$ | n（i）|/| i | $的$ h $的$ |/| i | $。
For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was introduced by Erdős and Galvin. Recently, the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimization problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of $H$, the number of components of $H$, and the expansion ratio $|N(I)|/|I|$ of the independent sets of $H$.
