Matching Tag: kneser-hypergraphs
Let K G r ( n , k ) KG^r(n,k) K G r ( n , k ) k k k [ n ] [n] [ n ] S ⊆ [ n ] S\subseteq[n] S ⊆ [ n ] r r r x , y ∈ S x,y\in S x , y ∈ S
Let [ n ] = { 1 , … , n } [n]=\{1,\ldots,n\} [ n ] = { 1 , … , n } K G r ( n , k ) s -stab \mathrm{KG}^r(n,k)_{s\textup{-stab}} KG r ( n , k ) s -stab r r r s s s k k k [ n ] [n] [ n ]
Let [ n ] = { 1 , … , n } [n]=\{1,\ldots,n\} [ n ] = { 1 , … , n } G \mathcal G G [ n ] [n] [ n ] r ≥ s ≥ 2 r\geq s\geq 2 r ≥ s ≥ 2 G { s -stable } ~ \mathcal G_{\widetilde{\{s\text{-stable}\}}} G { s -stable }
Let r ≥ 2 r\geq 2 r ≥ 2 F \mathcal{F} F F r − stab ~ \mathcal{F}_{\widetilde{r-\operatorname{stab}}} F r − stab
Let n ≥ 1 n\ge 1 n ≥ 1 r ≥ 2 r\ge 2 r ≥ 2 s ≥ r s\ge r s ≥ r F \cal F F [ n ] [n] [ n ] s s s F s \cal F_s F s KG r ( F s ) \operatorname{KG}^r(\cal F_s) KG r ( F s )
Let q q q P P P m m m V 1 , … , V m V_1,\dots,V_m V 1 , … , V m q − 1 q-1 q − 1 q q q
Let A ⊂ [ n ] A\subset[n] A ⊂ [ n ] K G A r ( n , k ) \mathrm{KG}^r_A(n,k) KG A r ( n , k ) r r r k k k σ ⊂ [ n ] \sigma\subset[n] σ ⊂ [ n ] σ ⊄ A \sigma\not\subset A σ ⊂ A
Let r ≥ 2 r\ge 2 r ≥ 2 k ≥ 1 k\ge 1 k ≥ 1 n ≥ r k n\ge rk n ≥ r k P = { P 1 , … , P ℓ } \mathcal P=\{P_1,\dots,P_\ell\} P = { P 1 , … , P ℓ } [ n ] [n] [ n ] ∣ P i ∣ ≤ r |P_i|\le r ∣ P i ∣ ≤ r i i i
For positive integers n , k , r n,k,r n , k , r k ≥ 2 k\geq 2 k ≥ 2 r ≥ 2 r\geq 2 r ≥ 2 n ≥ r k n\geq rk n ≥ r k K G r ( n , k ) r - s t a b KG^{r}(n,k)_{r\text{-}stab} K G r ( n , k ) r - s t ab r r r r r r K G r ( n , k ) KG^{r}(n,k) K G r ( n , k )
Ziegler–Alon–Drewnowski–Łuczak conjecture.
Let G = ( V , E ) \mathcal{G}=(V,E) G = ( V , E ) G ′ = ( V ′ , E ′ ) \mathcal{G}'=(V',E') G ′ = ( V ′ , E ′ ) G × G ′ \mathcal{G}\times\mathcal{G}' G × G ′ V × V ′ V\times V' V × V ′
