Quasi-Carousel Tournaments
Leonardo Nagami Coregliano

TL;DR
This paper characterizes sequences of large tournaments that are nearly balanced in outdegree and locally transitive, by analyzing the asymptotic behavior of specific sub-tournaments and their densities.
Contribution
It introduces new characterizations of almost balanced, asymptotically locally transitive tournament sequences using quasi-random properties.
Findings
Characterizations of tournament sequences with vanishing $W_4$ and $L_4$ densities.
Connections between local transitivity and quasi-random properties.
Conditions under which large tournaments approximate locally transitive structures.
Abstract
A tournament is called locally transitive if the outneighbourhood and the inneighbourhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments and , which are the only tournaments up to isomorphism on four vertices containing a unique -cycle. On the other hand, a sequence of tournaments with is called almost balanced if all but vertices of have outdegree . 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 and in goes to zero as goes to infinity.
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