论文标题
纯对。 X.比赛和强大的Erdos-Hajnal财产
Pure pairs. X. Tournaments and the strong Erdos-Hajnal property
论文作者
论文摘要
锦标赛中的纯对$ g $是订购的一对$（a，b）$ v（g）$的差异子集，因此$ b $中的每个顶点都与$ a $的每个顶点相邻。哪些锦标赛$ h $的财产是，如果$ g $是一项不包含$ h $作为子比赛的锦标赛，而$ | g |> 1 $，则有一个纯对$（a，b）$（a，b）$ g $，带有$ | a | | a |，| b | \ ge c | g | g | $，$ c | $，$ c> 0 $ c> $ c> 0 $是$ g $的常数独立于$ g $吗？让我们说这样的锦标赛$ h $具有强大的EH-Property。 据我们所知，可能是，当它的顶点套件具有线性订购时，锦标赛$ h $具有该物业，其后卫形成了森林。当然，这种情况是必要的，但我们远非证明足够的。我们朝这个方向迈出了一小步，表明如果可以最多可以与三个后卫一起订购比赛，那么它具有强大的EH-Property（除了一个案例，我们无法决定）。特别是，每个带有六个顶点的比赛都有该财产，除了我们无法决定的三场比赛。我们还提供了没有强大的EH-Property的七个vertex锦标赛。 这与Erdos-Hajnal的猜想有关，该猜想的一种形式说，对于每个锦标赛$ h $，存在$τ> 0 $，因此每个锦标赛$ g $不包含$ h $，因为子锦标赛具有至少$ | g |^τ$的频繁的基数子量。让我们说，满足此功能的比赛$ h $具有EH-Property。众所周知，与强大的EH-Property的每场比赛都有EH-Property。因此，我们的结果扩大了Berger，Choromanski和Chudnovsky的工作，他们证明了每场比赛最多六个顶点都有EH Property，除了他们没有决定的比赛。
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 $|A|,|B|\ge c|G|$, 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 Erdos-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.
