A note on vertex Turán problems in the Kneser cube

Abstract

The Kneser cube $K{n}_n$ has vertex set $2^{[n]}$ and two vertices $F,F^{\prime }$ are joined by an edge if and only if $F\cap F^{\prime }=\varnothing$. For a fixed graph $G$, we are interested in the most number $vex(n,G)$ of vertices of $K{n}_n$ that span a $G$-free subgraph in $K{n}_n$. We show that the asymptotics of $vex(n,G)$ 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 $2^{n-1}-vex(n,G)$ and $2^n-vex(n,G)$ in these two cases. In the case of bipartite $G$, we relate this problem to instances of the forbidden subposet problem.