Acyclic Sets in $k$-Majority Tournaments

Abstract

When $\Pi$ is a set of $k$ linear orders on a ground set $X$, and $k$ is odd, the $k$-majority tournament generated by $\Pi$ has vertex set $X$ and has an edge from $u$ to $v$ if and only if a majority of the orders in $\Pi$ rank $u$ before $v$. Let $f_k(n)$ be the minimum, over all $k$-majority tournaments with $n$ vertices, of the maximum order of an induced transitive subtournament. We prove that $f_3(n)\ge\sqrt{n}$ always and that $f_3(n)\le 2\sqrt{n}-1$ when $n$ is a perfect square. We also prove that $f_5(n) \ge n^{1/4}$. For general $k$, we prove that $n^{c_k} \le f_k(n) \le n^{d_k(n)}$, where $c_k = 3^{-(k-1)/2}$ and $d_k(n)\to \frac{1+\lg\lg k}{-1+\lg k}$ as $n\to \infty$.