THE SZEMERÉDI REGULARITY LEMMA AND ITS APPLICATION
YAQIAO LI
In this note we will prove Szemerédi’s regularity lemma, and its application in proving the
triangle removal lemma and the Roth’s theorem on 3AP.
1. The Regularity Lemma
Consider a bipartite graph given by vertex sets A, B, let E(A, B) be the set of edges between
them, we define the density of this bipartite graph (A, B) as
d(A, B) :=
|E(A, B)|
.
|A||B|
Definition 1 (-regular). Say the bipartite graph (A, B) is -regular if for any A0 ⊆ A, B 0 ⊆ B
with |A0 | ≥ |A| and |B 0 | ≥ |B|, we have
|d(A0 , B 0 ) − d(A, B)| ≤ .
Intuitively, this definition of regularity can be understood as the bipartite graph given by
A, B is random looking, or that the edges are uniformly distributed.
Let’s continue. We say a partition is equipartition if the number of elements in different parts
differ by at most one.
Definition 2 (-regular equipartition). Say an equipartition (of V (G), say) given by V1 , . . . , Vk
is -regular if all but at most k 2 of the pairs (Vi , Vj ) are -regular.
That is, most pairs of this partition give random looking bipartite graphs, with just a tiny
fraction of irregular parts. Alert! Although, that different pairs can have very different edge
densities (as bipartite graphs).
The Szemerédi’s regularity lemma guarantees the existence of regular equipartition of arbitrary graphs.
Theorem 1 (Szemerédi’s regularity). For every > 0, there exists T () > 0 (which depends
on only, not depend on specific graphs) such that every graph G has an -regular equipartition
into k parts, where the bounds of k is given as follows
1
≤ k ≤ T ().
The upper bound T () is a tower of 2.
2. Application: The Triangle Removal Lemma
We will use this regularity lemma to prove the triangle removal lemma.
Theorem 2 (Triangle removal lemma). For every > 0, there exists δ = δ() > 0 such that
if G is -far from being triangle free (that is, we have to remove at least n2 edges to make G
triangle free), then G contains at least δn3 triangles, where n is the number of vertices of G.
Date: April, 2015.
McGill University, [email protected].
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YAQIAO LI
If a graph contains Ω(n3 ) triangles, we then have to remove Ω(n2 ) edges to make it triangle
free since deleting one edge can remove at most n − 2 triangles. This lemma says the converse:
if G contains o(n3 ) triangles, then it can be made triangle free by deleting o(n2 ) edges.
We need the following lemma to prove this theorem.
Lemma 1. For any tri-partite graph given by vertex sets A, B, C such that
d(A, B) = b,
d(A, C) = c,
d(B, C) = a,
and
a ≥ 2, b ≥ 2, c ≥ 2,
if all these three pairs (A, B), (A, C) and (B, C) are -regular, then the number of triangles in
this tri-partite graph is at least
(1 − 2)(a − )(b − )(c − )|A||B||C|.
This lemma says that if a tri-partite graph is pairwise “dense” and regular, then it contains
a positive portion of triangles (i.e., a lot of triangles).
Proof. Let us consider those vertices in A having small neighbourhood in B,
S := {v ∈ A : |NB (v)| < (b − )|B|} ⊆ A,
where NB (v) denotes the set of neighbours of v in B. We will show the set S is very small,
specifically we claim
|S| < |A|.
Assume otherwise, then for the pair (S, B), we have that |S| ≥ |A| and |B| ≥ |B|, since (A, B)
is -regular, we have
|d(S, B) − d(A, B)| = |d(S, B) − b| ≤ .
But
|S| × (b − )|B|
|E(S, B)|
<
= b − ,
d(S, B) =
|S||B|
|S||B|
contradicting to the preceding inequality. Hence we have the size S is small as claimed.
Similarly, we have the set
T := {v ∈ A : |NC (v)| < (c − )|C|},
is also small, where NC (v) denotes the set of neighbours of v in C. Specifically, we have
|T | < |A|.
Now let us look at the remaining vertices of A, that is the set
A∗ := A − S − T,
then we know the set A∗ is large, specifically,
|A∗ | ≥ (1 − 2)|A|.
By our choice of S and T , we know that every vertex v ∈ A∗ has big neighbours both in B and
in C,
|NB (v)| ≥ (b − )|B|}, |NC (v)| ≥ (c − )|C|, ∀ v ∈ A∗ .
It follows by the fact that the pair (B, C) is -regular that
|d(NB (v), NC (v)) − d(B, C)| = |d(NB (v), NC (v)) − a| ≤ ,
∀ v ∈ A∗ .
In particular we have
d(NB (v), NC (v)) ≥ a − ,
implying that
|E(NB (v), NC (v))| ≥ (a − )|NB (v)||NC (v)| ≥ (a − )(b − )(c − )|B||C|.
THE SZEMERÉDI REGULARITY LEMMA AND ITS APPLICATION
3
Observe that |E(NB (v), NC (v))| is exactly the number of triangles lying in ({v}, NB (v), NC (v)).
Hence
#triangles in (A, B, C) ≥ #triangles in (A∗ , B, C)
≥ ((1 − 2)|A|) × ((a − )(b − )(c − )|B||C|) .
Now we prove the triangle removal lemma. The idea is to first use regularity lemma to give a
regular equipartition, then appropriately clean up the partition such that we can focus only on
those dense tri-partite graphs, and then apply the preceding lemma to conclude we have a lot
of triangles.
Proof. Step 1: Apply regularity lemma. Let’s apply regularity lemma with
G has an 4 -regular equipartition into k parts, where
4
to the graph G, then
4
≤ k ≤ T ( ).
4
Say this equipartition is given by V1 , . . . , Vk .
Step 2: Clean up. Think of this equipartition as an equipartition of the adjacency matrix of
G. We will clean up this partition by removing the following three “bad” parts: the diagonal,
the irregular parts, and the sparse parts. Fortunately, these parts are small enough, so we are
still left a large and dense parts to work with.
Remove the diagonal. That is, to remove all edges in every Vi , this will remove at most
(n/k)2
k 2 ≤ 18 n2 edges.
Remove the irregular parts. By the definition of 4 -regular equipartition, we have at most 4 k 2
irregular pairs, hence removes at most 4 k 2 (n/k)2 ≤ 41 n2 edges.
Remove the sparse parts. Since we are working with 4 -regularity, in the preceding lemma we
think a bipartite pair is “dense” if the edge density is larger than 2 4 = 2 . Hence we remove all
pairs which has edge density less than 2 , this will remove at most k2 2 (n/k)2 ≤ 14 n2 .
In total, we have removed at most 85 n2 edges.
Step 3: Apply the preceding lemma. Since G is -far from triangle free, we know that there are
still triangles in G. Suppose the tri-partite graph (Vi , Vj , Vl ) contains a triangle, then there is at
least one edge between each pair, hence the edges in every pair have not been removed in step
2. It follows that i, j, l are pairwise unequal and they are pairwise regular and dense. Applying
the preceding lemma, we have that (Vi , Vj , Vl ) contains at least
(1 − 2 )( − )3 (n/k)3 ≥ δn3 ,
4 2 4
triangles, in which
3
1
δ = (1 − )
,
2 64 T (/4)3
as desired.
Note that we have used both the lower bound and the upper bound from Szemerédi’s regularity
lemma to obtain a formula for δ in the triangle removal lemma. The sharp bound of δ is still
an open question.
3. Application: Roth’s 3AP Theorem
We now use the triangle removal lemma to show Roth’s theorem.
Theorem 3 (Roth). For any > 0, there exists N = N () > 0 such that for all n ≥ N , if
A ⊆ Zn satisfies |A| ≥ n, then A contains a nontrivial 3AP.
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YAQIAO LI
Proof. We will construct a graph G and apply the triangle removal lemma.
Let G be a tri-partite graph defined by vertex sets V1 = V2 = V3 = Zn , so |V (G)| = 3n.
Define edges for G as follows:
• For r ∈ V1 , s ∈ V2 , put (r, s) ∈ G if and only if s − r ∈ A;
• For s ∈ V2 , t ∈ V3 , put (s, t) ∈ G if and only if t − s ∈ A;
• For r ∈ V1 , t ∈ V3 , put (r, t) ∈ G if and only if (t − r)/2 ∈ A;
Consider (r, s, t) ∈ V1 × V2 × V3 and suppose they form a triangle, by our choice of edges, this
means that
s − r ∈ A, t − s ∈ A, (t − r)/2 ∈ A,
t−r
hence we get (s − r, 2 , t − s) is a 3AP in A. The problem is that this might be just a trivial
3AP, i.e., we may have s − r = t−r
2 = t − s = a ∈ A, then this “trivial” triangle is formed by
(s, s + a, s + 2a) ∈ V1 × V2 × V3 where s ∈ Zn , a ∈ A, that is, each trivial triangle corresponds
to a distinct pair (s, a) ∈ Zn × A, hence different trivial triangles are disjoint. We have
n2 ≤ #trivial triangles ≤ n2 .
The lowerbound implies that G is 9 -far away from being triangle free, by triangle removal
lemma this implies that we have at least δ(3n)3 triangles for some δ = δ() > 0. Now use the
upper bound for the trivial triangles we know that there must exist some nontrivial triangles,
equivalently nontrivial 3AP in A, as long as n is sufficiently large depending on .
4. Proof of the Regularity Lemma
At last, let us prove the regularity lemma. The proof is an “energy argument”. We will define
a notion of energy associated with each partition, we will see that lack of regularity implies
energy increment, but this energy is bounded above, hence the regularity must appear after a
finite steps of refining (of the partition).
Given a real matrix Am×n = (Aij ), define its energy as
X
E(A) :=
A2ij ≥ 0.
1≤i≤m
1≤j≤n
Obversely, if the entries of A are bounded by r, then a trivial upper bound is E(A) ≤ mnr2 .
2
In particular, if A is an n × n adjacency matrix of
P some graph G, then E(A) ≤ n .
Aij
be the density of A. Construct a new
For every matrix Am×n = (Aij ), let d(A) := i,j
mn
matrix Bm×n := (Bij ) where Bij = d(A) for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. This new matrix
can be viewed as a “smoothing” of the original matrix A. Let π be an operator that maps
every matrix A to its smoothing π(A) = B as just defined. Note that π(A) is a matrix with all
entries being a constant, hence the smoothing operation is unique. Conversely, A can be viewed
as a “mixing” of π(A). Note, however, that π(A) can have different mixings: for two matrices
A 6= B, it is possible that π(A) = π(B) = C as long as d(A) = d(B), hence A, B are smoothed
to the same matrix C, so either A or B can be viewed as a mixed version of C.
Here is a useful fact, it says that the energy decreases as we smooth a matrix, conversely, the
energy increases as we mix a matrix.
Lemma 2. For every matrix Am×n = (Aij ), we have
X
E(A) − E(π(A)) =
(Aij − d(A))2 = E(A − π(A)).
1≤i≤m
1≤j≤n
In particular, we have E(π(A)) ≤ E(A).
Proof. It’s a direct calculation.
THE SZEMERÉDI REGULARITY LEMMA AND ITS APPLICATION
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Let M be a partition of [m] and N be a partition of [n], for S ∈ M, T ∈ N , note that
S ⊆ [m], T ⊆ [n], let AS×T be the corresponding submatrix of A, hence M and N together
define a partition of matrix A. Let us denote this partition as PM,N := (M, N ), we also denote
PM,N := {AS×T : S ∈ M, T ∈ N }. Let us smooth every submatrix AS×T to π(AS×T ), and
denote this new blockwise-smoothed matrix of A to be PM,N (A). For example, in two extreme
cases, we have
• If M = {[m]} and N = {[n]}, i.e., the matrix is not partitioned, then PM,N (A) = π(A)
is just the smoothing of A;
• If M0 = {{1}, {2}, . . . , {m}} and N 0 = {{1}, {2}, . . . , {n}}, i.e., the matrix is partitioned
into singletons, then PM0 ,N 0 (A) = A since the smoothing of each singleton is just itself.
Remember that given a partition PM,N = (M, N ) of matrix A, the matrix PM,N (A) is a
blockwise-smoothed version of A, let its entry be denoted by pij , i.e., PM,N (A) = (pij ). Given
another partition PM0 ,N 0 = (M0 , N 0 ) that is a refinement of PM,N , let its entry be denoted by
p0ij , i.e., PM0 ,N 0 (A) = (p0ij ). A useful generalization of the above lemma is the following.
Lemma 3. For every matrix Am×n = (Aij ), and any two partitions PM,N and PM0 ,N 0 of matrix
A where PM0 ,N 0 is a refinement of PM,N , we have
X
E(PM0 ,N 0 (A)) − E(PM,N (A)) =
(p0ij − pij )2 = E(PM0 ,N 0 (A) − PM,N (A)).
1≤i≤m
1≤j≤n
In particular we have E(π(A)) ≤ E(PM,N (A)) ≤ E(PM0 ,N 0 (A)) ≤ E(A).
Proof. Just view each block(that is, a submatrix) of PM,N as an individual matrix, then a
refinement of PM,N is just a mixing of each block, then apply the previous lemma to each block.
Observe also that we have
π(PM,N (A)) = π(PM0 ,N 0 (A)) = π(A),
that is, PM,N (A), PM0 ,N 0 (A) and A all can be viewed as different mixing versions of π(A), the
difference is that they are successively refined mixing versions.
Definition 3 (-regularity of a matrix). Say a matrix Am×n is -regular if for any S ⊆ [m], T ⊆
[n] with |S| ≥ m, |T | ≥ n, we have |d(AS×T ) − d(A)| ≤ .
The following is important.
Lemma 4 (Lack of Regularity implies bounded energy increment). Suppose Am×n is not regular, then there is a partition PM,N of A such that
E(PM,N (A)) − E(π(A)) > mn4 .
Proof. As A is not -regular, there exists S ⊆ [m], T ⊆ [n] with |S| ≥ m, |T | ≥ n such that
|d(AS×T ) − d(A)| > .
Define a partition PM,N := (M, N ) as M = {S, [m] − S} and N = {T, [n] − T }. Apply lemma
3 we have
X
E(PM,N (A)) − E(π(A)) =
(pij − d(A))2
1≤i≤m
1≤j≤n
≥
X
(pij − d(A))2 =
i∈S,j∈T
2
X
(d(AS×T ) − d(A))2
i∈S,j∈T
4
> |S||T | ≥ mn ,
as desired.
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YAQIAO LI
Definition 4 (-regular partition of a matrix). Say a partition PM,N of a matrix Am×n is
-regular if all except at most |M||N | pairs (S, T ) ∈ M × N satisfy that AS×T is -regular.
As a corollary of Lemma 4 we have the following.
Lemma 5 (Lack of Regularity implies bounded energy increment-2). Suppose an equipartition
PM,N of a matrix Am×n is not -regular, then there is a refined equipartition PM0 ,N 0 such that
E(PM0 ,N 0 (A)) − E(PM,N (A)) > mn5 .
Proof. Assume |M| = k, |N | = l, that is, the rows and columns of A are partitioned equally into
k and l parts, respectively. As PM,N is not -regular, there exist at least kl pairs (S, T ) ∈ M×N
such that AS×T are not -regular. By Lemma 4, we can partition each block AS×T such that this
4
0
0
block has energy increment > |S||T |4 = mn
kl . Define the refined partition PM ,N of the whole
matrix A to be the “intersection” of the partitions of each block, if it is not an equipartition,
refine it appropriately to make it into an equipartition. Observe that the effect of PM0 ,N 0 on
each block is an even finer partition than the one used to achieve bounded energy increment, by
Lemma 3, the energy can only increase even more. Hence by Lemma 3 and Lemma 4 we have
mn 4
) = mn5 ,
E(PM0 ,N 0 (A)) − E(PM,N (A)) > (kl) · (
kl
as desired.
We have seen that refine the irregular partition induces bounded energy increment, but the
energy is trivially bounded above by mnr if all the entries are bounded by r, hence the refining
process must terminate, meaning that the regularity of partition must be achieved after a finite
number of steps.
Now we can give a sketch of the proof of the regularity lemma.
Proof. Let An×n be the adjacency matrix of a graph G. Start with k = 1 -equipartition of A,
repeat the refinement process to achieve the regularity (of partition) in a finite number of steps,
specifically, in at most n2 /(n2 5 ) = 1/5 steps. It should be noted that to apply Lemma 5, we
should also take care of the “size” of the refinement equipartition, i.e., it cannot be too large.
It can be shown that from a k-equipartition PM,N , one can take an at most 8k -equipartition
PM0 ,N 0 to achieve the energy increment bound in Lemma 5. Hence eventually we will reach an
-regular l-equipartition where l is bounded above by a tower of the form 8k in at most 1/5
levels.