Let T1, . . . , Tc be the components of T . Since each Ti is isomorphic to a subgraph
of G1, for every i ∈ [c] we have |E(Ti)|/(|V (Ti)| − 1) ≤ m1(G1). In particular,
|S| =
c
∑
i=1
|E(Ti)| ≤ m1(G1)
c
∑
i=1
(|V (Ti)| − 1) = m1(G1)(v − c),
so (v − c)/|S| ≥ 1/m1(G1). Therefore
μ′({F ⊆ G2 : E(F) ⊇ S}) ≤ Cm1(G1)
1
n
!|S|(v−c)/|S|
≤
C1
n1/m1(G1)
|S|
,
as desired.
End of proof of Lemma 23.9.
Proof of Theorem 23.7 We randomly partition V (G) into random clusters U1, . . . ,Um
where |V (F)| | |Ui| and |Ui| ∼ C log n for i = 1, 2, . . . , m. The degree of v ∈ Ui in
G[Ui] dominates Bin((δF +α)n, (|Ui|−1)/(n−1)) and so if C is sufficiently large,
then w.h.p. δ (Ui) ≥ (δF + α/2)|Ui| for i = 1, 2, . . . , m. So, w.h.p. each Ui con-
tains an F-factor. A simple calculation shows that for every set of distinct vertices
x1, . . . , xs ∈ V (H) and every function f : [s] → [m],
Pr xi ∈ Uf (i) for each i ∈ [s] ≤
C log n
n
s
. (23.23)
Theorem 23.7 from Lemma 23.9 (with C replaced by C log n) and Theorem 23.1.
this is the kinda stuff that truly scares me. i dont want to sit and decipher what you are trying to explain, i want to understand what you are trying to explain. i wish there was a resource that explained absolutely complex ideas like these in the simplest of words
this is the kinda stuff that truly scares me. i dont want to sit and decipher what you are trying to explain, i want to understand what you are trying to explain. i wish there was a resource that explained absolutely complex ideas like these in the simplest of words
> open PDF file
> 698 pages
jesus