PROVING SZEMERÉDI’S REGULARITY LEMMA
GEON LEE
Abstract. This paper reviews the most common approach to the proof of
Szemerédi’s regularity lemma, as presented by R. Diestel [1].
1
We only consider simple graphs in this paper. A graph G = (V, E) is a relational
structure with the non-empty vertex set V and the edge set E ⊆ V 2 . |G| is the
order or the number of vertices in G. kGk is the total number of edges in G. For
disjoint vertex sets C and D, kC, Dk is the number of edges between C and D. A
partition
P = {V1 , V2 , · · · , Vk } of V is the set of nonempty subsets in V such that
S
V
=
V
, and Vi ∩ Vj = ∅ for all i 6= j. Any partition P 0 of V is a refinement of
i
i
P if every element of P 0 is a subset of some element of P.
2
Szemerédi’s regularity lemma is an immensely powerful tool in extremal graph theory. Endre Szemerédi introduced the weaker version of the lemma to prove the
Erdös-Turán conjecture (1936) that any sequence of natural numbers with positive
density contains a (long) arithmetic progression. He proved the full lemma in his
later paper (1978). Roughly speaking, regularity lemma states that the vertex set
of any (large) graph can be divided into subsets of equal size such that edges between the subsets are distributed randomly; we expect to find an edge connecting
two such subsets with some probabilty p, just as in a randomly generated graph.
The lemma thereby lets us apply some well-known properties of the random graph
to all classes of graph.
In order to precisely state the regularity lemma, we require the following definitions. For two disjoint sets of vertices, C and D, define
d(C, D) :=
kC, Dk
|C||D|
as the density of (C, D). The notion of regularity extends from here.
For some > 0, a pair of sets (C, D) is -regular if for all subsets X ⊆ C and
Y ⊆ D with |X| > |C|, |Y | > |D|, we have
|d(C, D) − d(X, Y )| 6 A partition P = {V0 , V1 , · · · , Vk } of V is -regular if the following holds:
(1) |V0 | < |V |
(2) |V1 | = |V2 | = · · · = |Vk |
(3) All but at most k 2 pairs (Vi , Vj ), 1 ≤ i < j ≤ k, are -regular.
1
2
GEON LEE
Note that V0 is ignored when determining the regularity of the partition. V0 is
the exceptional set, which allows all the other sets in P to be of equal size. In
the proof of the regularity lemma, we will repeat the process of splitting the graph
into smaller and smaller subsets until we obtain the desired partition; |V0 | can be
thought of as a bin that contains all vertices that are left over from the splitting
operations.
Regularity lemma simply states that any graph with at least some number of
vertices contains an -regular partition:
Theorem 2.1 (Szemerédi’s regularity lemma). For every > 0 and every integer
m ≥ 1, there exists an integer M such that any graph G = (V, E) of order at least
m admits an -regular partition P = {V0 , V1 , · · · , Vk }, with m ≤ k ≤ M .
The proof of the theorem heavily relies on the potential function, f , which will
be defined shortly. The function takes a partition (or sets of vertices) as input and
returns a positive real number that measures ‘how regular’ the input is. f is defined
as the following: for two disjoint vertex sets C and D in a graph of order n,
|C||D| 2
d (C, D)
f (C, D) :=
n2
kC, Dk2
= 2
n |C||D|
S
S
For two different partitions C = i Ci and D = i Di ,
X
f (C, D) :=
f (Ci , Dj )
i,j
For a partition P =
S
i Vi ,
f (P) :=
X
f (Vi , Vj )
i<j
If a partition P = {V0 , V1 , · · · , Vk } contains an exceptional set V0 , then consider V0
as the set of singletons:
X
X
f (P) :=
f (Vi , Vj ) +
f ({v}, Vi )
16i<j
i>1
where v ∈ V0 .
Also note that the function f is bounded above by 1:
X
f (P) =
f (Vi , Vj )
i<j
=
X |Vi ||Vj |
i<j
≤
n2
· d2 (Vi , Vj )
X |Vi ||Vj |
i<j
n2
≤1
Given a partition with enough non--regular pairs in it, our main goal is to construct its refinement that is substantially ‘more regular’ than before (as measured
by f ). Then we may choose any arbitrary partition in G with its sets of equal size
except the exceptional set; and refine it again and again to obtain a sequence of
PROVING SZEMERÉDI’S REGULARITY LEMMA
3
partitions with bigger f values. However only bounded number of such refinements
is possible until f reaches its upper bound. Hence the sequence cannot continue
infinitely, and we expect to obtain a partition that satisfies the conditions of the
theorem some time along this process. The detailed proof follows in the next section.
3
P
P 2P 2
bi , we note some useUsing the Cauchy-Schwartz inequality, (ai bi )2 6
ai
ful properties of the S
potential function
f
.
When
two
disjoint
sets C and D are
S
partitioned into C = Ci and D = Di respectively, f always increases:
X kCi , Dj k2
n2 |Ci ||Dj |
i,j
P
( i,j kCi , Dj k)2
P
> 2P
|Ci | |Dj |
n
= f (C, D)
f (C, D) =
S
S
Also, if a partition P = Ci is refined to P 0 = Ci , where Ci is the set of
subsets of Ci , then f increases as well:
f (P) =
X
f (Ci , Cj )
i<j
6
X
f (Ci , Cj )
i<j
6 f (P 0 )
since f (P 0 ) =
P
i<j
f (Ci , Cj ) +
P
i
f (Ci )
Now we shall see how far f may increase for a pair of two vertex sets if each set
is partitioned into two subsets. The increment is yet less than any constant.
Lemma 3.1. Let C and D be disjoint sets of vertices in a graph of order n. If the
pair (C, D) is not -regular, then for all > 0, there exist partitions C = {C1 , C2 }
and D = {D1 , D2 } such that
f (C, D) > f (C, D) + 4
|C||D|
n2
Proof. Fix . Since (C, D) is -regular, there exist subsets C1 ⊆ C and D1 ⊆ D,
with |C1 | > |C| and |D1 | > |D|, and |d(C1 , D1 ) − d(C, D)| > . Let C2 :=
C \ C1 , D2 := D \ D1 ; and C := {C1 , C2 }, D := {D1 , D2 }. Lastly, define δ :=
d(C1 , D1 ) − d(C, D). It follows that
kC1 , D1 k =
kC, Dk|C1 ||D1 |
+ δ|C1 ||D1 |
|C||D|
4
GEON LEE
Via the Cauchy-Schwartz inequality and the above equation,
n2 f (C, D) =
X kCi , Dj k2
i,j
|Ci ||Dj |
X kCi , Dj k2
kC1 , D1 k2
+
|C1 ||D1 |
|Ci ||Dj |
i+j>2
P
( i+j>2 kCi , Dj k)2
kC1 , D1 k2
>
+ P
|C1 ||D1 |
i+j>2 |Ci ||Dj |
=
kC1 , D1 k2
(kC, Dk − kC1 , D1 k)2
+
|C1 ||D1 |
|C||D| − |C1 ||D1 |
2
|C1 |2 |D1 |2
kC, Dk
+ δ 2 |C1 ||D1 | +
=
|C||D|
kC, Dk − kC1 , D1 k
2
kC, Dk
>
+ δ 2 |C1 ||D1 |
|C||D|
=
> n2 f (C, D) + 4 |C||D|
Using the above results, we will show that given a non--regular partition, there
exists its refinement that increases f by some constant.
Lemma 3.2. 0 < 6 1/3. Let P = {V0 , V1 , · · · , Vk } be a partition of a graph G
of order n, with |V0 | 6 n and |V1 | = · · · = |Vk | := d. If P is not -regular, there
exists another partition P 0 = {V00 , V10 , · · · , Vl0 } of G such that the following holds:
(i) k 6 l 6 k4k
(ii) |V00 | 6 |V0 | + n/2k
(iii) |V10 | = · · · = |Vl0 |
(iv) f (P 0 ) > f (P) + 5 /2
Proof. For 1 6 i 6 j 6 k, define partitions Vij of Vi and Vji of Vj as follows: if the
pair (Vi , Vj ) is not -regular, let Vij := {Vi } and Vji := {Vj }; if not, then we can
find partitions Vij of Vi and Vji of Vj , with |Vij | = |Vji | = 2 and
f (Vij , Vji ) > f (Vi , Vj ) + 4 d2 /n2 .
Consider two elements in Vi as equivalent whenever they are in the same set in
Vij for all j 6= i, and let Vi be the set of such equivalence classes. Then V := {V0 } ∪
Sk
i=1 Vi is a partition of G that refines P. Note that each Vi may be partitioned
into two sets at most k − 1 times. Due to our construction of Vi , we have at most
2k−1 equivalance classes in it; hence |Vi | 6 2k−1 , and k 6 |V| 6 k2k .
PROVING SZEMERÉDI’S REGULARITY LEMMA
5
Since Vi , Vj each refines Vij and Vji ,
X
X
X
f (V) =
f (Vi ) +
f (V0 , Vi ) +
f (Vi , Vj )
i
> f (V0 ) +
X
i>1
16i<j
f (V0 , {Vi }) +
X
i>1
> f (V0 ) +
X
f (Vij , Vji )
16i<j
f (V0 , {Vi }) +
i>1
X
f (Vi , Vj ) + k 2
16i<j
4 d 2
n2
k 2 d2
= f (P) + 5 2
n
5
> f (P) +
2
We shall once again refine V by cutting each Vi into sets of equal size (that is
much smaller than before) and adding all the remaining elements to the exceptional
set V0 . The subsets must be small enough so that we do not obtain too many
‘remainder’ elements during the refinement; thereby the resulting exceptional set
|V00 | is kept small, or less than |V |. To obtain the desired partition, let {V10 , · · · , Vl0 }
be the maximal collection of disjoint sets of size d0 = bd/4k c, with each Vi0 contained
Sl
entirely within some set V ∈ Vj , where j > 1. Lastly let V00 := V \ i=1 Vi0 ; and
P 0 := {V00 , V10 , · · · , Vl0 }. Then, due to the choice of d0 ,
k 6 l 6 k4k
and
|V00 | 6 |V0 | + d0 |V|
d
(k2k )
4k
dk
= |V0 | + k
2
n
6 |V0 | + k
2
6 |V0 | +
Finally, since P 0 refines V, we have
f (P 0 ) > f (V) > f (P) +
5
2
The proof of the regularity lemma easily follows from the repeated application
of Lemma 3.2.
Proof of the regularity lemma. Fix m > 1 and > 0. Let G(V, E) of order
n > m be given. Also assume that 6 1/3 with no loss of generality. Before
applying the above lemma to G, we must (i) require that the size of the exceptional
set is bounded above by n at all times; and (ii) determine M .
(i) f increases by at least 5 /2 per each iteration of Lemma 3.2, which therefore
may be repeated at most 2/5 times before f hits the upper bound of 1. To obtain
an -regular partition with the size of its exceptional set less than or equal to n,
6
GEON LEE
choose k > m large enough so that
2 n
n
· k 6
5
2
2
or
2k−2 > 1/6
holds. Next, we require that the exceptional set in the initial partition does not
exceed n/2, by only considering n large enough. For |V0 | < k, we would like to
have k 6 n/2, or n > 2k/. Note that for n small, the regularity lemma will
produce trivial results. (See the final paragraph of the proof.)
(ii) Given the initial partition with k sets in it, the number of sets may grow up
to k4k after applying Lemma 3.2. Let g is the map that takes k to k · 4k . Then
we may choose M as:
2k
2/5
M := max g
(k),
Finally we are ready to construct an -regular partition {V0 , V1 , · · · , Vk } of G.
If n 6 M , let k := n and V0 = ∅, so that the given graph is partitioned into k
singletons. Otherwise, let V0 be the minimal set such that k divides |V \ V0 |; hence
|V0 | < k 6 n/2. Now place the remaining vertices into sets of any equal size, and
apply Lemma 3.2 repeatedly until the desired partition is obtained.
Acknowledgements. I wish to express my indebtedness to my mentors, John
Wilmes and John Wiltshire-Gordon.
References
[1] R. Diestel. Graph Theory. Springer. 2010.
[2] A. Marcus. The Szémeredi Regularity Lemma. 2008.
http://cs-www.cs.yale.edu/homes/marcus/papers/srl.pdf