Consider finite directed graphs with no loops and no multiple directed edges, but allowing both \(\overrightarrow{xy}\) and \(\overrightarrow{yx}\).
A monotone tournament is an orientation of a complete graph for which some ordering of the vertices directs every edge from the smaller vertex to the larger. A complete digraph has both directed edges between every two distinct vertices.
Given \(m\ge1\), prove that for all sufficiently large \(N\), every digraph on \(N\) vertices has an \(m\)-vertex subset inducing either an empty digraph, a complete digraph, or a monotone tournament.