The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e. one with edge density either very close to 0 or 1) of size $\mathrm{\Omega }(n)$. We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size $\mathrm{\Omega }(\log n)$. An example of Rödl from 1986 shows that this bound is tight.

Let $R_r(t)$ denote the size of the largest non-universal r-graph $\mathcal{G}$ so that neither $\mathcal{G}$ nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erdős–Hajnal-type stepping-up lemma, showing how to transform a lower bound for $R_r(t)$ into a lower bound for $R_{r+1}(t)$. As an application of this lemma, we improve a bound of Conlon–Fox–Sudakov by showing that $R_3(t)\ge t^{\mathrm{\Omega }(t)}$.

Erdős-Hajnal定理断言，非通用图，即不包含某些固定图H的归纳副本的图，具有均匀的大小集，其大小显着大于通常期望在图中找到的大小。我们在r-一致超图的设置中获得了这种味道的两个结果。

Rödl定理断言，如果一个n顶点图是非通用的，则它包含几乎同构的大小（即边缘密度非常接近0或1的集合）$\mathrm{\Omega }（ñ）$。我们证明如果3一致的超图是非通用的，则它包含几乎同构的大小集$\mathrm{\Omega }（\mathrm{日志}ñ）$。1986年罗德（Rödl）的例子表明，这个界限很严格。

让 ${\mathrm{[R}}_{\mathrm{[R}}（Ť）$表示最大的非通用r图的大小$\mathcal{G}$ 这样既不 $\mathcal{G}$它的补码也没有包含大小为t的完整r- partite子图。我们证明了一个Erdős–Hajnal型逐步提升引理，展示了如何变换${\mathrm{[R}}_{\mathrm{[R}}（Ť）$ 进入下界 ${\mathrm{[R}}_{\mathrm{[R}+\mathrm{1个}}（Ť）$。作为该引理的一种应用，我们通过证明以下几点改进了Conlon–Fox–Sudakov的边界${\mathrm{[R}}_3（Ť）\ge {Ť}^{\mathrm{\Omega }（Ť）}$。