The Kneser cube \(Kn_n\) has vertex set \(2^{[n]}\) and two vertices \(F,F'\) are joined by an edge if and only if \(F\cap F'=\emptyset \) . For a fixed graph G, we are interested in the most number \(\textrm{vex}(n,G)\) of vertices of \(Kn_n\) that span a G-free subgraph in \(Kn_n\) . We show that the asymptotics of \(\textrm{vex}(n,G)\) is \((1+o(1))2^{n-1}\) for bipartite G and \((1-o(1))2^n\) for graphs with chromatic number at least 3. We also obtain results on the order of magnitude of \(2^{n-1}-\!\textrm{vex}(n,G)\) and \(2^n-\!\textrm{vex}(n,G)\) in these two cases. In the case of bipartite G, we relate this problem to instances of the forbidden subposet problem.
