A note on vertex Turán problems in the Kneser cube

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