Acyclic Subgraphs in $k$-Majority Tournaments

TL;DR

This paper improves the upper bound on the size of the largest acyclic subgraph in 3-majority tournaments, refining our understanding of their structure in voting models.

Contribution

It provides a tighter upper bound on the maximum size of transitive sub-tournaments in 3-majority tournaments, advancing previous bounds.

Findings

Improved upper bound: $f_3(n) < \,\sqrt{2n} + 0.5$

Enhanced understanding of acyclic subgraphs in majority tournaments

Refined theoretical limits for voting scenario models

Abstract

A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph is used to model voting scenarios, where the vertices correspond to options ranked by $k$ voters. When $k$ is odd, the resulting digraph is always a tournament, called $k$-majority tournament. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive sub-tournament. Recently, Milans, Schreiber, and West proved that $\sqrt n \le f_3(n) \le 2 \sqrt n +1 $. In this paper, we improve the upper bound of $f_3(n)$ by showing that $f_3(n) < \sqrt {2n} +\frac 12 $.