Description
The next meeting of the seminar on Extremal Sets will be on April 11
at 12:15-13:45 in the Nagyterem of Renyi Institute.
Speaker: Daniel Gerbner
Title: Vertex Turán problems in the Kneser cube
(joint work with Balázs Patkós)
Abstract: 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 $\vex(n,G)$ of vertices of $Kn_n$ that span a $G$-free subgraph in $Kn_n$. 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.