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.
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We would like to thank Jan Volec for checking many small cases using a computer and Igor Balla for drawing our attention to . We are also extremely grateful to the anonymous referees for their careful reading of the paper and many useful suggestions and comments.
Supported in part by ISF Grant 1028/16 and ERC Starting Grant 633509.
Research supported in part by SNSF grant 200021 196965.
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Bucić, M., Long, E., Shapira, A. et al. Tournament Quasirandomness from Local Counting. Combinatorica 41, 175–208 (2021). https://doi.org/10.1007/s00493-020-4371-y
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