Erdös’ conjecture on multiciplities of
complete subgraphs revisited
(lower upper bound for cliques of
size 5 and 6)
F. Franek, McMaster University,
Hamilton, Ontario, Canada
kt(G) = # of cliques of order t in a
graph G
kt(n) = min{kt(G)+kt(Ĝ):|G|=n}
Ĝ denotes the complement of G
kt(n)
et(n) =
n
(t )
et = lim et(n)
n

CS-2001 (Sedmihorky)
Slide 1
et minimum proportion of
monochromatic Kt’s in a edgecolouring of Kn with two colours.
An old (1962) conjecture of Erdös
related to Ramsey theory states that
et
t
1= 2 (2 )
It follows from Goodman’s (1959)
work that the conjecture is true for t
= 3. Erdös & Moon (1964) showed
that the conjecture modified for
bipartite cliques is true. Sidorenko
(1980’s) showed that the conjecture
modified for cycles is true, while
not true for certain incomplete
subgraphs.
CS-2001 (Sedmihorky)
Slide 2
The conjecture is true for random
and “pseudo-random” graphs
(Graham & Spencer 1971, Frankl,
Rödl, & Wilson 1988, Thomason
1985). Franek & Rödl 1993 showed
that the conjecture holds not only
for “pseudo-random” graphs, but
for graphs obtained from “pseudorandom” graphs by small
“perturbations”. Thomason showed
the original conjecture false for t  4
in 1989. He obtained following
upper bounds:
e4 < 0.976 x 2-5
e5 < 0.906 x 2-9
CS-2001 (Sedmihorky)
Slide 3
et < 0.936 x
t
21-( 2 )
for t > 5
As for the lower bound Giraud
1
1979 showed that e4 > 46
Franek & Rödl 1993 used simple
Cayley graphs together with a
computer search to disapprove the
conjecture for t = 4 with virtually the
same upper bound as Thomason.
Searches for higher values for t were
not computationally feasible at that
time.
In this paper we will present the
same technique for t = 5 and t = 6
obtaining better upper bounds that
Thomason:
CS-2001 (Sedmihorky)
Slide 4
e5 < 0.886 x 2-9 < 0.906 x 2-9
e6 < 0.745 x 2-14 < 0.936 x 2-14
The method:
For a fixed graph G = (V, S)
Gn = (Vn,Sn) is defined in the
following way - every vertex v
from V is “blown up” to av, a
set of size n. Every two
elements of av are joined by an
edge; x av and y aw are
joined by an edge iff v and w are
joined by an edge in G.
CS-2001 (Sedmihorky)
Slide 5
G
Gn
Now we can calculate the number of
k-cliques in Gn from number of mcliques in G.
3
2
n
2
( )( )
n
n n
2 (3 ) nk (G) + (2 )(2 ) k (G) +
n
(4) k (G)
k4(Gn) = n4k4(G) +
2
n2k3(G) +
2
1
k4(Ĝn) = n4k4(Ĝ)
CS-2001 (Sedmihorky)
Slide 6
n
2
( ) n k (G) +
n n
n
3 (2 )(2 ) nk (G) + 3 (2 ) n k (G) +
n n
n
2 (3 )(2 ) k (G) + 2 (4 ) nk (G) +
n
(5) k (G)
n5k
k5(Gn) =
5(G) + 4
3
2
3
2
4
3
2
1
k5(Ĝn) = n5k5(Ĝ)
n6k
k6(Gn) =
n
3
6(G) + 5
n
4
n
2
( ) n k (G) +
4
5
( ) (G) + 3 ( ) n k (G) +
n
n n
2 (5 ) nk (G) + 2 (4 )(2 ) k (G) +
4
n3k
4
2
2
3
n
3
n
2
3
2
n
3
n
3
( ) 2 ( ) nk (G) + ( )( ) k (G) +
3
2
CS-2001 (Sedmihorky)
Slide 7
4
2
n
2
n
2
n
2
n
2
n
2
( )( )( ) n k (G) +
2
4
n
6
( )( )( ) k (G) + ( ) k (G)
3
1
k6(Ĝn) = n6k6(Ĝ)
This leads immediately to the
following limit formulas
lim
n

k4(Gn)+k4(Ĝn)
=
|Gn|
4
( )
24(k4(G)+k4(Ĝ))+36k3(G)+14k2(G)+
k1(G) over k1(G)4
CS-2001 (Sedmihorky)
Slide 8
k5(Gn)+k5(Ĝn)
lim

n
=
|Gn|
5
( )
120(k5(G)+k5(Ĝ))+240k4(G)+
150k3(G)+30k2(G)+k1(G) over k1(G)5
lim
n

k6(Gn)+k6(Ĝn)
=
|Gn|
6
( )
720(k6(G)+k6(Ĝ))+1800k5(G)+
1560k4(G)+540k3(G)+62k2(G)+k1(G)
over k1(G)6
CS-2001 (Sedmihorky)
Slide 9
Consider a Cayley graph GX,E :
vertices are subsets of a set X,
E X}, a family of
sizes.
a, b X are joined by an edge
iff |a D b| E
a D b denotes the symmetric
difference.
How to count cliques in GX,E :
<a0,a1,...,am-1> is an X,E,msequence iff each ai is a subset
of X, |ai| X, and |ai D aj| E
whenever i j
cm(X,E) denotes the set of all
X,E,m-sequences.
CS-2001 (Sedmihorky)
Slide 10
Observation:
2 |X| c (X,E)
km+1(GX,E) =
m
(m+1)!
Thus
k4((GX,E)n)+k4((ĜX,E)n)
lim
=
n 
|(GX,E)n|
4
c3(X,E)+c3(X,Ê)+6c2(X,E)+
7c1(X,E)+1 over 23|X|
(
)
k5((GX,E)n)+k5((ĜX,E)n)
lim
=
n 
|(GX,E)n|
5
(
)
CS-2001 (Sedmihorky)
Slide 11
c4(X,E)+c4(X,Ê)+10c3(X,E)+
25c2(X,E)+15c1(X,E)+1 over 24|X|
k6((GX,E)n)+k6((ĜX,E)n)
lim
=
n 
|(GX,E)n|
6
(
)
c5(X,E)+c5(X,Ê)+15c4(X,E)+
65c3(X,E)+90c2(X,E)+31c1(X,E)+1
over 25|X|
Sequences can be calculated by
generating them and tallying
them along the process. Since
done by a computer program,
must be careful not to miss any.
CS-2001 (Sedmihorky)
Slide 12
Illustration of the method
a0
y0
y02
y01
y012
y2
y1
a1
y12
a2
<a0,a1,a2> is an X,E,3-sequence iff
|y0| + |y01| + |y02| + |y012| E
|y1| + |y01| + |y12| + |y012| E
|y2| + |y02| + |y12| + |y012| E
|y0| + |y01| + |y2| + |y12| E
|y0| + |y02| + |y1| + |y01| E
|y1| + |y01| + |y2| + |y02| E
CS-2001 (Sedmihorky)
Slide 13
Generate all sequences
<m0,m1,m2,m01,m02,m12,m012> so that
m0 + m01 + m02 + m012 E
m1 + m01 + m12 + m012 E
m2 + m02 + m12 + m012 E
m0 + m01 + m2 + m12 E
m0 + m02 + m1 + m01 E
m1 + m01 + m2 + m02 E
for each sequence calculate product
|X| 
|X|-m0 
|X|-m0-m1 
|X|-m0-m1-m2

m0 
m1 
m2
m01


|X|-m0-m1 -m2 -m01 
|X|-m0-m1 -m2 -m01 -m02

m02
m12


|X|-m0-m1 -m2 -m01 -m02 -m12

m012

CS-2001 (Sedmihorky)
Slide 14
For X = {1,2,3,4,5,6,7,8,9,10}
E = {1,3,4,7,8,10}, Ê = {2,5,6,9}
we calculated
c5(E) = 13677741000
c5(Ê) = 25382760480
c4(E) = 742203000
c4(Ê) = 1009617840
c3(E) = 14734170
c3(Ê) = 17273850
c2(E) = 125730
c1(E) = 506
Thus
e4  0.97650119 x 2-5
e5  0.88583369880 x 2-9
e6  0.744513803200 x 2-14
CS-2001 (Sedmihorky)
Slide 15