6.896 Sublinear Time Algorithms
October 7, 2004
Lecture 8
Lecturer: Ronitt Rubinfeld
Scribe: Kayi Lee
1
Review
In the previous lecture, we introduced the Szemer´
edi’s Regularity Lemma, which states
that all graphs can be approximated by randomlooking graphs in a certain sense.
Definition 1
A pair of disjoint vertex sets
(
A, B
)
is
γ

regular
if, for all
A
⊆
A, B
⊆
B
such that

A
 ≥
γ

A

and

B
 ≥
γ

B

, we have

d
(
A, B
)
−
d
(
A , B
)

< γ
, where
d
(
A, B
) =
e
(
A,B
)

A

B

is the density between the sets
A
and
B
.
Note that if (
A, B
) is
γ
regular, (
A, B
) is also
γ
regular for all
γ
≤
γ
≤
1 by
definition.
Lemma 2 (Szemer´
edi’s Regularity Lemma)
For all
m,
>
0
, there exists
T
such
that if
G
= (
V, E
)
with

V

> T
and
A
is a equipartition of
V
into
m
sets, then there
exists a equipartition
B
, a refinement of
A
, with
k
sets such that:
1.
m
≤
k
≤
T
; and
2. there are fewer than
k
2
pairs of set partitions that are not
regular
We also proved the following lemma, which states that a triplet of pairwise regular
vertex sets behaves like a random graph with respect to the number of distinct triangles
in the graph.
Lemma 3 (Komlos and Simonovits)
For all
η >
0
, there exists
γ
Δ
and
δ
Δ
such that
if
A, B, C
are disjoint subsets of
V
and each pair is
γ
Δ
regular with density at least
η
,
then
G
contains at least
δ
Δ

A

B

C

distinct triangles with vertex from each of
A, B
and
C
.
Note that the constants
T, γ
Δ
and
δ
Δ
in the above two lemmas depend on the choices
of
m,
and
η
, but not the size of the graph.
In the rest of the note, we will denote
them as as
T
(
m,
)
, γ
Δ
(
η
) and
δ
Δ
(
η
) to remind readers of the parameters these constants
depend on.
1