SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 645-667, March 2024.
Abstract. Fix [math], and let [math] be a graph, with vertex set partitioned into [math] subsets (“blocks”) of approximately equal size. An induced subgraph of [math] is “transversal” (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly [math] vertices). A “pure pair” in [math] is a pair [math] of disjoint subsets of [math] such that either all edges between [math] are present or none are; and in the present context we are interested in pure pairs [math] where each of [math] is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.

中文翻译：

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纯对。九．横向树

SIAM 离散数学杂志，第 38 卷，第 1 期，第 645-667 页，2024 年 3 月。
摘要。修复 [math]，并让 [math] 成为一个图，其中顶点集被划分为大小大致相等的 [math] 子集（“块”）。如果 [math] 的导出子图在每个块中恰好有一个顶点（因此它恰好有 [math] 顶点），则它是“横向的”（相对于此分区）。 [math] 中的“纯对”是 [math] 的不相交子集的一对 [math]，使得 [math] 之间的所有边都存在或不存在；在当前上下文中，我们对纯对 [math] 感兴趣，其中每个 [math] 都是其中一个块的子集，而不是同一个块。本文收集了一些结果和开放性问题，涉及如果排除各种类型的横向子图，必须存在多大的纯对。