  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 $.
