Pure pairs. X. Tournaments and the strong Erdos-Hajnal property
Pure pairs. X. Tournaments and the strong Erdos-Hajnal property
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纯对。
DOI：
10.1016/j.ejc.2023.103786
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发表时间：
2024
影响因子：
1
通讯作者：
Chudnovsky M
中科院分区：
文献类型：
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作者：
Chudnovsky M
竞赛图G中的纯对是V(G)的不相交子集的有序对(A，B)，使得B中的每个顶点都与A中的每个顶点相邻。哪些竞赛图H具有这样的性质：如果G是不包含H作为子竞赛图的竞赛图，且|G|>；1，则G中存在一个纯对(A，B)且|A|，|B|≥c|G|，其中c>；0是独立于G的常数？假设这样的竞赛图H具有强EH-性质。据我们所知，竞赛图H具有这一性质的充要条件是它的顶点集有一个线性序，在这个序中，它的分支构成一个森林。当然，这一条件是必要的，但我们远不能证明这是充分的。我们朝这个方向迈出了一小步，证明了如果一个锦标赛可以用至多三个巴贝克排序，那么它就具有强EH性质(除了我们无法决定的一种情况)。特别是，除了我们不能决定的三个顶点外，每一个至多有六个顶点的锦标赛都有这个性质。我们还给出了一个不具有强EH性质的七点竞赛图。这与ERDőS-哈杰纳尔猜想有关，该猜想的一种形式是对每个竞赛图H都存在τ>；0使得每个不包含H作为子竞赛图的竞赛图G至少有一个基数传递子竞赛图|G|τ。假设满足这一条件的竞赛图H具有EH-性质。众所周知，每个具有强EH-性质的竞赛图也具有EH-性质，因此我们的结果推广了Berger，Choromanski和Chudnovsky的工作，他们证明了每个至多有6个顶点的竞赛图都有EH-性质，除了一个他们没有决定的竞赛。
A pure pair in a tournament G is an ordered pair (A, B) of disjoint subsets of V (G) such that every vertex in B is adjacent from every vertex in A. Which tournaments H have the property that if G is a tournament not containing H as a subtournament, and| G|> 1, there is a pure pair (A, B) in G with| A|,| B|≥ c| G|, where c> 0 is a constant independent of G? Let us say that such a tournament H has the strong EH-property. As far as we know, it might be that a tournament H has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdős-Hajnal conjecture, which in one form says that for every tournament H there exists τ> 0 such that every tournament G not containing H as a subtournament has a transitive subtournament of cardinality at least| G| τ. Let us say that a tournament H satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide.