Quasi‐carousel tournaments

Abstract

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.

On the Maximum Density of Fixed Strongly Connected Subtournaments

“…We remark that (1) partially confirms a conjecture proposed by the first author in [12]. Finally, the last homomorphism studied is what we call here triangular homomorphism φ C 3 .…”

“…First we recall the definition of the carousel homomorphism φ R ∈ Hom + (A 0 , R) as the limit of the sequence (R 2n+1 ) n∈N of carousel tournaments. Analogously to quasi-random properties, the quasi-carousel properties [12] are equivalent properties over a homomorphism φ ∈ Hom + (A 0 , R) that force φ = φ R . We recall two of the carousel properties below.…”

“…The theory of quasi-randomness started with the study of quasi-random graphs in the seminal papers by Thomason [31] and Chung, Graham and Wilson [9] and now has branches in several other theories such as uniform hypergraphs [7,6,4], graph orientations [19], permutations [11,22] and tournaments [8,21,12]. We do not attempt to provide a detailed review of the theory of quasi-randomness here (see [23] for a survey), but its main gist is that there are several a priori different properties of a homomorphism φ ∈ Hom + (A 0 , R), called quasi-random properties, that force φ = φ qr .…”

Flag algebras and tournaments

“…Theorem 3.2.2 (C. [Cor15,Theorem 3.1]). If F ∈ F A 3 is an A-flag of size 3 and G is either O A + I A or C A 3 + Tr A 3 , then…”

“…Corollary 3.3.3 (C. [Cor15,Corollary 3.4]). If φ ∈ Hom + (A 0 , R), then φ(R 4 ) 1/2 with equality if and only if φ = φ R .…”