Quasi‐carousel tournaments

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  W 4 and  L 4 , which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments  ( T n ) n ∈ Nwith  V ( T n ) = n is called almost balanced if all but  o ( n ) vertices of  T n 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  W 4 and  L 4 in  T n goes to zero as  n goes to infinity.