On clique growth in products of directed graphs (Q1385297)

The Sperner product of digraphs \(G_1=(V_1, E_1)\) and \(G_2=(V_2, E_2)\) is the digraph \(G_1 \times G_2\), the vertex set of which is the Cartesian product \(V= V_1 \times V_2\) of the vertex sets of the two, and there is an arc in \(G_1 \times G_2\) pointing from \((a,b)\) to \((a',b')\) if either or both of the relations \((a,a')\in E_1\) and \((b,b')\in E_2\) hold. The paper studies the growth of the largest transitively oriented clique in the Sperner powers of a fixed graph and derives the first nontrivial upper bounds on the Sperner capacity of arbitrary digraphs.