A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on \(n\ge 2\) vertices that does not contain H or its complement as an induced subgraph has a pure pair of size \(\Omega (n)\); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size \(n^{1-c}\), where \(0<c<1\). Let H be a graph: does every graph on \(n\ge 2\) vertices that does not contain H or its complement as an induced subgraph have a pure pair of size \(\Omega (|G|^{1-c})\)? The answer is related to the congestion of H, the maximum of \(1-(|J|-1)/|E(J)|\) over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and \(\overline{H}\). We show that the answer to the question above is “yes” if \(d\le c/(9+15c)\), and “no” if \(d>c\).

图G中大小为t的纯对是V ( G ) 的不相交子集的一对A 、 B ，每个基数至少为t ，使得A对于B来说要么是完备的，要么是反完备的。众所周知，对于每个森林H ，在\(n\ge 2\)个顶点上不包含H或其补集作为导出子图的每个图都有一对大小为\(\Omega (n)\)的纯对；此外，只有当H或其补集是森林时，这才成立。在本文中，我们研究大小为\(n^{1-c}\)的纯对，其中\(0<c<1\) 。设H为图：在\(n\ge 2\)个顶点上不包含H或其补集作为导出子图的每个图是否都具有一对大小为\(\Omega (|G|^{1-c })\) ?答案与H的拥塞有关，即H的所有带有边的子图J上的最大值\(1-(|J|-1)/|E(J)|\) 。 （拥塞是非负的，当H是森林时，拥塞恰好等于零。）令d为H和\(\overline{H}\)的拥塞中较小的一个。我们证明，如果\(d\le c/(9+15c)\) 则上述问题的答案为“是”，如果\(d>c\) 则答案为“否”。