On the Density of Transitive Tournaments

Coregliano, Leonardo Nagami; Razborov, Alexander A.

doi:10.1002/jgt.22044

Cited by 34 publications

(48 citation statements)

References 20 publications

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“…Finally, let us prove that 6 ( , 1∕2) implies 1 . If satisfies 6 ( , 1∕2), we have already proved that it must be balanced (since [ ( )] = 1∕2) and from equality assertion of (8) and (9) and the fact that the second moment of a (0, 1)-random variable is 1∕3, we have that ( 4 ) ⩾ 1∕2, hence = by Corollary 3.4. ■…”

Section: +1mentioning

confidence: 87%

“…Types and flags used the other two classes are from [9]). This can be stated formally and cleanly 4 by saying that if ⟨ , ⟩ is a random arc of picked uniformly at random, then the sequence of random variables ( ( , )) ∈ℕ converges almost surely to 1∕4.…”

Section: F I G U R Ementioning

confidence: 99%

“…Later, Griffiths [15] proved that (Tr 4 ) is minimized if and only if is the quasi-random homomorphism qr . Finally, the minimization problem for a single tournament was closed when Griffith's result was extended in [9]: for ⩾ 4, the density (Tr ) is minimized if and only if is the quasi-random homomorphism qr . Now, if we consider the analogous maximization problem, the gears completely reverse: the transitive case becomes the trivial case (since (Tr , Tr ) = 1 for every ⩽ ) and property 2 of Chung and Graham says that ( ⃗ 3 ) is maximized if and only if is the limit of an almost balanced sequence.…”

Section: Introductionmentioning

confidence: 99%

“…Later, Griffiths proved that

φ ({Tr}_{4})

is minimized if and only if ϕ is the quasi‐random homomorphism

φ_{qr}

. Finally, the minimization problem for a single tournament was closed when Griffith’s result was extended in : for

k ⩾ 4

, the density

φ ({Tr}_{k})

is minimized if and only if ϕ is the quasi‐random homomorphism

φ_{qr}

.…”

mentioning

confidence: 99%

“…) divided by

| V (T) | - 2

. A set of interesting characterizations of quasi‐randomness says that if F is any of

O^{A}

I^{A}

{Tr}_{3}^{A}

, or

{\vec{C}}_{3}^{A}

, then a sequence of tournaments

{false(T_{n} false)}_{n \in double-struckN}

is quasi‐random if and only if

F (u, v)

is “nearly” 1/4 for “almost all” arcs

⟨ u, v ⟩

(the theorems for

O^{A}

and

I^{A}

are from and the theorems for the other two classes are from ). This can be stated formally and cleanly by saying that if

⟨ {boldu}_{boldn}, {boldv}_{boldn} ⟩

is a random arc of

T_{n}

picked uniformly at random, then the sequence of random variables

{false(F ({bold-italicu}_{boldn}, {bold-italicv}_{boldn}) false)}_{n \in double-struckN}

converges almost surely to 1/4.…”

mentioning

confidence: 99%

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Quasi‐carousel tournaments

Coregliano

2017

Journal of Graph Theory

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A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W4 and L4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments false(Tnfalse)n∈double-struckN with Vfalse(Tnfalse)=n is called almost balanced if all but o(n) vertices of Tn have outdegree (1/2+o(1))n. In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W4 and L4 in Tn goes to zero as n goes to infinity.

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Section: +1mentioning

confidence: 87%

Section: F I G U R Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Later, Griffiths proved that

φ ({Tr}_{4})

is minimized if and only if ϕ is the quasi‐random homomorphism

φ_{qr}

. Finally, the minimization problem for a single tournament was closed when Griffith’s result was extended in : for

k ⩾ 4

, the density

φ ({Tr}_{k})

is minimized if and only if ϕ is the quasi‐random homomorphism

φ_{qr}

.…”

mentioning

confidence: 99%

“…) divided by

| V (T) | - 2

. A set of interesting characterizations of quasi‐randomness says that if F is any of

O^{A}

I^{A}

{Tr}_{3}^{A}

, or

{\vec{C}}_{3}^{A}

, then a sequence of tournaments

{false(T_{n} false)}_{n \in double-struckN}

is quasi‐random if and only if

F (u, v)

is “nearly” 1/4 for “almost all” arcs

⟨ u, v ⟩

(the theorems for

O^{A}

and

I^{A}

are from and the theorems for the other two classes are from ). This can be stated formally and cleanly by saying that if

⟨ {boldu}_{boldn}, {boldv}_{boldn} ⟩

is a random arc of

T_{n}

picked uniformly at random, then the sequence of random variables

{false(F ({bold-italicu}_{boldn}, {bold-italicv}_{boldn}) false)}_{n \in double-struckN}

converges almost surely to 1/4.…”

mentioning

confidence: 99%

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Quasi‐carousel tournaments

Coregliano

2017

Journal of Graph Theory

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Characterization of quasirandom permutations by a pattern sum

Chan

Kráľ

Noel

et al. 2020

Random Struct Algorithms

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It is known that a sequence {Π i } i∈N of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Π i converges to 1∕24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Π i converges to |S|∕24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.

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