  We study a high-dimensional analog for the notion of an acyclic (aka
transitive) tournament. We give upper and lower bounds on the number of
$d$-dimensional $n$-vertex acyclic tournaments. In addition, we prove that
every $n$-vertex $d$-dimensional tournament contains an acyclic subtournament
of $\Omega(\log^{1/d}n)$ vertices and the bound is tight. This statement for
tournaments (i.e., the case $d=1$) is a well-known fact. We indicate a
connection between acyclic high-dimensional tournaments and Ramsey numbers of
hypergraphs. We investigate as well the inter-relations among various other
notions of acyclicity in high-dimensional to tournaments. These include
combinatorial, geometric and topological concepts.
