A note on vertex Turán problems in the Kneser cube
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911779972448256 |
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| author | Gerbner, Dániel Patkós, Balázs |
| author_facet | Gerbner, Dániel Patkós, Balázs |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_02525 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on vertex Turán problems in the Kneser cube Gerbner, Dániel Patkós, Balázs Combinatorics 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. |
| title | A note on vertex Turán problems in the Kneser cube |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.02525 |