Tournament Quasirandomness from Local Counting

Abstract

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.

Quasirandom Latin squares

“…In other words, the quasirandomness of graphs is captured by densities of two (small) subgraphs. Results of a similar kind have been obtained for many other types of combinatorial objects, for example groups [17], hypergraphs [4,15,16,19,21], permutations [3,9,10,22], which also appear in disguise in statistics [1,13,20,26], set systems [5], subsets of integers [7] and tournaments [2,6,12,18]. In this paper, we prove a conjecture posed by Garbe, Hancock, Hladký and Sharifzadeh [14,Conjecture 12.3] and establish that the same phenomenon holds for Latin squares.…”

“…The integral in (8) is the probability that if points x 1 , x 2 , y 1 and y 2 are chosen randomly from Ω, numbers z ab randomly from W (x a , y b ) for a, b ∈ [2], and a number z randomly uniformly from [0, 1], then z ab ≤ z for all a, b ∈ [2]. Observe that when x 1 , x 2 , y 1 and y 2 are fixed, if we choose y 3 randomly from Ω and z 13 randomly from W (x 1 , y 3 ), then the number z 13 is randomly uniformly chosen from [0, 1] by (2).…”

“…The integral in (8) is the probability that if points x 1 , x 2 , y 1 and y 2 are chosen randomly from Ω, numbers z ab randomly from W (x a , y b ) for a, b ∈ [2], and a number z randomly uniformly from [0, 1], then z ab ≤ z for all a, b ∈ [2]. Observe that when x 1 , x 2 , y 1 and y 2 are fixed, if we choose y 3 randomly from Ω and z 13 randomly from W (x 1 , y 3 ), then the number z 13 is randomly uniformly chosen from [0, 1] by (2). Hence, the integral in ( 8) is the probability that if points x 1 , x 2 , y 1 , y 2 and y 3 are chosen randomly from Ω, numbers z ab randomly from W (x a , y b ) for a ∈ [2] and b ∈ [3], then z ab ≤ z 13 for all a, b ∈ [2].…”

“…Observe that when x 1 , x 2 , y 1 and y 2 are fixed, if we choose y 3 randomly from Ω and z 13 randomly from W (x 1 , y 3 ), then the number z 13 is randomly uniformly chosen from [0, 1] by (2). Hence, the integral in ( 8) is the probability that if points x 1 , x 2 , y 1 , y 2 and y 3 are chosen randomly from Ω, numbers z ab randomly from W (x a , y b ) for a ∈ [2] and b ∈ [3], then z ab ≤ z 13 for all a, b ∈ [2]. This probability is equal to the sum of the terms α A t(A, W, f ) over all 2 × 3 patterns A, where α A = 1/3 if the pattern A contains 5 and 6 in the same column and α A = 1/6 otherwise.…”

Lower bound on the size of a quasirandom forcing set of permutations

“…Graph quasirandomness is captured by several seemingly different but in fact equivalent conditions: the density of all subgraphs is close to their expected density in a random graph, all but the largest eigenvalue of the adjacency matrix are small, the density of a graph is uniformly distributed amongst its (linear size) subsets of vertices, all cuts between linear size subsets of vertices have the same density, etc. Besides graphs, there are results on quasirandomness of many different types of combinatorial structures, in particular, tournaments [2,6,13,18], hypergraphs [4,15,16,19,22], set systems [5], groups [17], subsets of integers [7], and Latin squares [10]. In this paper, we are concerned with quasirandomness of permutations as studied in [3,11,23].…”