A Further Extension of Rödl's Theorem

[摘要] 0$ and a nonnull graph $H$. A well-known theorem of Rödl from the 80s says that every graph $G$ with no induced copy of $H$ contains a linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces a subgraph with maximum degree at most $\varepsilon |S|$ in $G$ or its complement. There are two extensions of this result:0$" depending on $H$ and $\varepsilon$; and 0$ depending on $H$ and $\varepsilon$ such that $G$ is $(N,\varepsilon)$-restricted, which means $V(G)$ has a partition into at most $N$ subsets that are $\varepsilon$-restricted.0$, $\kappa$ and $N$ still exist so that for every $d\ge0$, every graph with at most $\kappa d^{\vert H\vert}$ induced copies of $H$ has an $(N,\varepsilon)$-restricted induced subgraph on at least $\vert G\vert-d$ vertices. This unifies the two aforementioned theorems, and is optimal up to$\kappa$ and $N$ for every value of $d$.

[学科分类] 统计和概率