论文标题
稀疏图的功率
Erdős-Hajnal properties for powers of sparse graphs
论文作者
论文摘要
我们证明，每一个无处一类的图表$ \ MATHCAL {C} $，正整数$ d $和$ \ VAREPSILON> 0 $，以下内容：每一个$ n $ vertex graph $ g $ from $ \ nathcal {c} $一个人都可以找到两个discoint $ a | （1/2- \ varepsilon）\ cdot n $和$ | b | =ω（n^{1- \ varepsilon}）$，以及$ \ peripatorName {dist}（a，a，b）\ leq d $ for a $ in a $ in $ n in $，b $，ob，$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a}（for $ b）和$ b \ in B $。我们还展示了该语句的一些更强的变体，包括对无处浓密图类的一阶解释的设置的概括。
We prove that for every nowhere dense class of graphs $\mathcal{C}$, positive integer $d$, and $\varepsilon>0$, the following holds: in every $n$-vertex graph $G$ from $\mathcal{C}$ one can find two disjoint vertex subsets $A,B\subseteq V(G)$ such that $|A|\geq (1/2-\varepsilon)\cdot n$ and $|B|=Ω(n^{1-\varepsilon})$ and either $\operatorname{dist}(a,b)\leq d$ for all $a\in A$ and $b\in B$, or $\operatorname{dist}(a,b)>d$ for all $a\in A$ and $b\in B$. We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dense graph classes.
