A “pure pair” in a graph G is a pair A, B of disjoint subsets of V(G) such that A is complete or anticomplete to B. Jacob Fox showed that for all \(\varepsilon >0\), there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with \(|A|,|B|\ge \varepsilon n\). He also proved that for all \(c>0\) there exists \(\varepsilon >0\) such that for every comparability graph G with \(n>1\) vertices, there is a pure pair A, B with \(|A|,|B|\ge \varepsilon n^{1-c}\); and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all \(c>0\), and all \(\ell _1, \ell _2\ge 4/c+9\), there exists \(\varepsilon >0\) such that, if G is an \((n>1)\)-vertex graph with no hole of length exactly \(\ell _1\) and no antihole of length exactly \(\ell _2\), then there is a pure pair A, B in G with \(|A|\ge \varepsilon n\) and \(|B|\ge \varepsilon n^{1-c}\). This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph.

图G中的“纯对”是V ( G )的不相交子集的一对A、  B，使得A对于B是完备的或反完备的。Jacob Fox 证明，对于所有\(\varepsilon >0\)，存在一个具有n 个顶点的可比图G ，其中n很大，其中不存在纯对A、  B与\(|A|,|B| \ge \varepsilon n\)。他还证明，对于所有\(c>0\)都存在\(\varepsilon >0\)，使得对于每个可比图G对于\(n>1\)个顶点，存在纯对A ,  B ，其中\(|A|,|B|\ge \varepsilon n^{1-c}\)；并推测这同样适用于每个完美图G。我们证明了这个猜想，并通过多种方式强化了它。特别是，我们证明对于所有\(c>0\)和所有\(\ell _1, \ell _2\ge 4/c+9\)，存在\(\varepsilon >0\)这样，如果G是一个\((n>1)\)顶点图，没有长度正好为\(\ell _1\)的孔且没有长度正好为\(\ell _2\)的反孔，则存在纯对A ,  B在G中，具有\(|A|\ge \varepsilon n\)和\(|B|\ge \varepsilon n^{1-c}\)。这进一步得到加强，通过排除一般图的一些“长”细分来取代排除孔。