Year: 2026
Type: article
Abstract: 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 ,
Source: Graphs and Combinatorics
Authors: Dániel Gerbner, Balázs Patkós
Institution: HUN-REN Alfréd Rényi Institute of Mathematics
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
OpenAlex ID: W7126239359