SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 281-293, January 2021.
A graph $G$ is Ramsey for a graph $H$ if every 2-coloring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a “sparse” graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that in general the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree.

中文翻译：

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有界树宽图的大小 Ramsey 数

SIAM 离散数学杂志，第 35 卷，第 1 期，第 281-293 页，2021 年 1 月。
如果 $G$ 的边的每 2 种着色都包含 $H$ 的单色副本，则图 $G$ 是图 $H$ 的拉姆齐。我们考虑以下问题：如果 $H$ 具有有界树宽，是否存在一个“稀疏”图 $G$ 是 $H$ 的 Ramsey？考虑了稀疏性的两个概念。首先，我们证明如果 $H$ 的最大度数和树宽是有界的，那么存在 $O(|V(H)|)$ 边的图 $G$ 是 $H$ 的 Ramsey。这以前仅因具有有限带宽的较小类别的图 $H$ 而为人所知。另一方面，我们证明，一般来说，对于 $H$ 是 Ramsey 的图 $G$ 的树宽不能仅根据 $H$ 的树宽而有界。事实上，即使树宽被退化替换并且$H$是一棵树，后一种说法也是正确的。