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{\bf Kevin G. Milans, Daniel H. Schreiber and Douglas B. West}
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{\bf Acyclic Sets in $k$-Majority Tournaments}
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When $\Pi$ is a set of $k$ linear orders on a ground set $X$, and $k$ is odd,
the \emph{$k$-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 $n\to \infty$.



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