COMBINATORICS AND GRAPH THEORY, P. 2 OF HW. III
2. (a) Assume that X1 , X2 , X3 are disjoint vertex sets in G with x1 , x2 , x3 vertices respectively. Assume furthermore that the graphs (Xi , Xj ), 1 ≤ i < j ≤ 3 are all 14 ε2 -regular with
density dij ≥ 2ε respectively. Let N denote the number of copies of K23 in G (the complete
tripartite graph with two vertices on each part). Show that
N ≥ c(x1 x2 x3 )2 ,
where c is a constant depending on ε, and x1 , x2 , x3 are sufficiently large.
(b) Deduce from here the Erdős-Stone theorem for H = K23 .
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Some incomplete proof sketch, you are invited to complete the detail.
(a) Let u1 , u2 , . . . , u6 be the vertices of H = K23 and U1 , U2 , U3 be the three parts of H. For
1 ≤ h ≤ 6, let
Lh := {u1 , . . . , uh }.
For each u ∈ Uj /Lh , let Tuh be the set of vertices in Xj adjacent to every element of
f (N (u) ∩ Lh ). The main idea is to find, by induction, many embeddings f of Lh into G
such that, for each u ∈ V (H)\Lh we always have
|Tuh | ≥ ε|N (u)∩Lh | |Xj |.
(1)
(You are invited to fill in the gap why this implies our result (a), especially when u = u6 .)
For h = 0 there is nothing to prove. We may therefore assume that Lh has been embedded
consistent with the induction hypothesis and attempt to embed u = uh+1 ∈ Uk into many
appropriate v ∈ Tuh .
Let Y be the set of neighbors of u which are not yet embedded.
Claim 0.1. The number of v ∈ Tuh \f (Lh ) such that for all y ∈ Y, |N (v) ∩ Tyh | ≥ ε|Tyh | is at
least 21 ε2 |Xj |.
By this claim, taking f (u) = v for any such v and Tyh+1 = N (v) ∩ Tyh will complete (1).
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COMBINATORICS AND GRAPH THEORY, P. 2 OF HW. III
Proof. (of the Claim) Let By be the set of vertices v in Tuh which are bad for y ∈ Y , that is
|N (v) ∩ Tyh | < ε|Tyh |.
Note that by induction, if y ∈ Ul for some 1 ≤ l ≤ 3 then |Tyh | ≥ ε2 |Xl |. Therefore, we must
have
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|By | < ε2 |Xj |,
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for otherwise the density between By and Tyh would be less than ε, contradicting the regularity assumption on G. Hence,
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|Tuh \ ∪y∈Y By | > ε2 |Xj | − 2 ε2 |Xj | = ε2 |Xj |.
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(b) We want to show that if G contains at least ( 41 + ε)n2 edges, then it contains a copy of
H. To do this we apply the method to prove the triangle removal lemma: first apply the
regularity lemma to obtain an equipartition (V0 , V1 , . . . , Vm ), and then apply the deletion
process: that is we delete edges
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between non ε0 -regular pairs;
between ε-singular pairs of density smaller than d0 ;
between V0 and any Vi ;
within each Vi .
By choosing ε0 and d0 carefully, we can guarantee that the number of deleted edges is at
most εn2 /2, and thus the new graph G0 has at least (1/4 + /2)n2 edges. Now we apply
Turán’s theorem, which implies that G0 contains a triangle xyz. Assume without loss of
generality that x ∈ V1 , y ∈ V2 , z ∈ V3 . It thus follows from (a) that there are many copies
of H in the tripartite graph (V1 , V2 , V3 ).