A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the ‘correct number’ of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T. We consider two types of counts, the global one and the local one.

We first observe that if T has the correct global count of H and h ≥ 7 then quasirandomness of T is only forced if H is transitive. The next natural question when studying quasirandom objects asks whether possessing the correct local counts of H is enough to force quasirandomness of T. A tournament H is said to be locally forcing if it has this property.

Variants of the local forcing problem have been studied before in both the graph and hypergraph settings. Perhaps the closest analogue of our problem was considered by Simonovits and Sós who looked at whether having ‘correct counts’ of a fixed graph H as an induced subgraph of G implies G must be quasirandom, in an appropriate sense. They proved that this is indeed the case when H is regular and conjectured that it holds for all H (except the path on 3 vertices). Contrary to the Simonovits-Sós conjecture, in the tournament setting we prove that a constant proportion of all tournaments are not locally forcing. In fact, any locally forcing tournament must itself be strongly quasirandom. On the other hand, unlike the global forcing case, we construct infinite families of non-transitive locally forcing tournaments.

仲和Graham的一个众所周知的定理指出，如果ħ ≥4然后锦标赛Ť当且仅当是准随机Ť包含每个ħ -点比赛的次为一subtournament“正确号码”。在本文中，我们调查的quasirandomness之间的关系ŧ和的计数单^ h -点锦标赛^ h的牛逼。我们考虑两种类型的计数，全局计数和局部计数。

我们首先注意观察，如果牛逼具有正确全球的H计和^ h ≥7则quasirandomness牛逼只有被迫如果^ h是传递的。在研究准随机对象时，下一个自然问题是，拥有H的正确局部数是否足以强制T的准随机性。如果锦标赛H具有此属性，则它是在本地强制执行的。

以前在图形和超图设置中都已经研究了局部强制问题的变体。Simonovits和Sós也许考虑了与我们问题最接近的类似物，他们研究了在适当的意义上将固定图H的“正确计数”作为G的诱导子图是否暗示G必须是准随机的。他们证明，当H是规则的并且猜想对于所有H都成立时，确实如此。（3个顶点上的路径除外）。与Simonovits-Sós的猜想相反，在锦标赛设置中，我们证明了所有锦标赛中有固定比例的不是本地强迫的。实际上，任何在本地进行的比赛本身都必须是严格准随机的。另一方面，与全局强迫情况不同，我们构造了无限个非传递性本地强迫比赛族。