On the maximum density of fixed strongly connected subtournaments

TL;DR

This paper investigates the maximum possible density of specific strongly connected 5-vertex subtournaments within large tournaments, identifying extremal configurations and characterizing certain recursive blow-up structures.

Contribution

It determines the asymptotic maximum densities for five fixed strongly connected 5-vertex subtournaments and characterizes tournaments avoiding specific configurations.

Findings

Identified maximum densities for five fixed strongly connected 5-vertex subtournaments.

Characterized tournaments as recursive blow-ups of a 3-cycle based on avoidance of certain 5-vertex tournaments.

Provided unique extremal sequences for each considered tournament.

Abstract

We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5.