Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ .This bound is known to be tight for $r=1$ . The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$ , as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$ .Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.

中文翻译：

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可比性图的改进的 Ramsey 类型结果

几个离散几何问题等价于估计图中的最大齐次集的大小，这些集恰好是少数几个的并集可比性图. 对此类结果的一个重要观察是，如果G是一个n- 顶点图，它是r可比性（或更一般地说，完美）图，然后要么G或其补集包含一个大小集团$n^{1/(r+1)}$.这个界限被认为是紧的$r=1$. 是否最适合的问题$r\ge 2$由 Dumitrescu 和 Tóth 研究。我们证明它本质上是最好的$r=2$，以及：我们引入了两个可比图的概率构造n顶点，其并集不包含团或独立的大小集$n^{1/3+o(1)}$.使用类似的思路，我们也可以构造图G那是联合r可比性图，两者都不是G其补集也不包含完整的二部图，其部分大小为$cn/{(log n)^r}$. 有了这个，我们改进了 Fox 和 Pach 的结果。