EVERY MONOTONE GRAPH PROPERTY HAS A SHARP

THRESHOLD

EHUD FRIEDGUT AND GIL KALAI

Abstract In their seminal work which initiated random graph theory Erdos and

Renyi discovered that many graph prop erties have sharp thresholds as the number

of vertices tends to innityWeprove a conjecture of Linial that every monotone

graph prop erty has a sharp threshold This follows from the following theorem

n

Let V pf g denote the Hamming space endowed with the probability

n

k nk

measure dened by p p where k

p p  n  n

e Let A b e a monotone subset of V Wesay that A is symmetric if there is a transitiv

n

permutation group on f ng suchthatA is invariant under

Theorem For every symmetric monotone Aif A then A

p q

for q p c log log nc is an absolute constant

Graph properties

A graph property is a prop erty of graphs which dep ends only on their isomorphism

class Let P b e a monotone graph prop erty that is if a graph G satises P then

every graph H on the same set of vertices whichcontains G as a subgraph satises

P as well Examples of such prop erties are G is connected G is Hamiltonian G

contains a clique complete subgraph of size t G is not planar the clique number

of G is larger than that of its complement the diameter of G is at most s etc

For a prop erty P of graphs with a xed set of n vertices we will denote by P

p

the probability that a random graph on n vertices with edge probability p satises

P The theory of random graphs was founded byErdos and Renyi and one of

their signicant discoveries was the existence of sharp thresholds for various graph

prop erties that is the transition from a prop erty b eing very unlikely to it b eing

very likely is very swift Many results on various asp ects of this phenomenon have

app eared since then In what follows c c etc are universal constants

Theorem Let P be any monotone property of graphs on n vertices If P

p

q p c log log n then P for

q

Research supp orted in part bygrants from the Israeli Academy of Sciences BSF the Sloan

foundation and bya grant from the state of Niedersachsen

EHUD FRIEDGUT AND GIL KALAI

This result veries a conjecture of Nati Linial and complements a theorem of

Bollobas and Thomason who proved that P for q c p

q

Let P b e the prop ertyG contains a clique with k nvertices and let k n log n

The threshold interval namely the interval of edgeprobabilities p where

p

in this case is of size prop ortional to log n Perhaps some words of

explanation are in order Consider G Gn p that is G is a random graph with

n vertices and edge probability p The length of the interval of probabilities p for

which the clique numb er size of maximal clique of G is almost surely k where

k log n is of order log n The transition b etween clique numbers k and k

o ccurs along an interval of length log n and this is precisely the threshold interval

of interest to us At the b eginning of this transition p erio d the probabilityofhaving

a clique of size k is This probability rises to at the end of this interval but

the probabilityofhaving a k clique is still small log n The value of p

must increase by c log n b efore the probability for having a k clique reaches

and another transition interval b egins

Conjecture Let P be any monotone property of graphs on n vertices If P

p

for q p c log log n then P

q

Symmetric properties

n

Consider the Hamming space V p f g endowed with the probability mea

n

k nk

sure dened by p p where k Put

p p n n

n

A f g V p Let A b e a monotone subset of f g that is if

n p

and for every i i n then A Wesaythat A is

i i n

symmetric if there is a transitive p ermutation group on n f ng such

that A is invariant under

Theorem For every symmetric monotone Aif A then A

p q

for q p c log log n

To deduce Theorem from Theorem with c c note that the family

of edgesets of graphs on n vertices which satisfy a monotone graph prop erty P

is invariant under the action of S the group of p ermutations of the vertices on

n

the edges Note that Theorem remains true for monotone prop erties of random

subgraphs of arbitrary nite edgetransitive graphs In particular the theorem applies

k

to random subgraphs of the discrete cub e and of C the pro duct of k copies of an

n

ncycle grids on the k dimensional torus

n

Note that the symmetry assumption is needed If A f f g

n

g then Ap Bollobas and Thomason proved using the Kruskal

p

Katona theorem that for every monotone A symmetricornotif A then

p

A for q cp q

SHARP THRESHOLDS FOR GRAPH PROPERTIES

In the rest of the section we will deduce Theorem via a Lemma of Margulis

and Russo see also Ch from a result of Bourgain Kahn Kalai

Katznelson and Linial briey BKKKL In the next section we will indicate a

few extensions of the BKKKL Theorem which imply several extensions of Theorem

In Section we will give examples showing that our results are tight In Section

we will discuss the connection of the threshold interval to the symmetry group

and in Section we will discuss some connections to results by MargulisTalagrand

Russo and KruskalKatona

P

For v A let hv jfw A distv w gj Dene A v hv

p p

v A

Lemma Margulis Russo

d A

p

Ap

p

dp

Next we need the notion of inuence Let X b e a probability space and let

n

f X f g b e a measurable map Dene the inuence of the kth variable

n

on f denoted by I k as follows For u u u u X set l u

f n k

fu u u tu u t X g and dene

k k n

n

u I k Pru X f is not constant on l

f k

Given a monotone set A V p denote by the characteristic function of AWe

n A

will write I k for I k Let A b e the sum of the inuences of the variables

A p

A

on Note that A Ap

A p p

Theorem Bourgain Kahn Kalai Katznelson and Linial For every func

n

tion f X f g with Prf t there is a variable k so that

log n

I k c t

f

n

where t mint t

Pro of of Theorem Since A is symmetric the inuence of eachvariable is

the same and therefore by Theorem the sum of the inuences for A in V r isat

n

least c t log nwheret min A A So by Lemma for every r such

r r

that A wehave d Adr c Alog n Therefore dlog Adr

r r r r

log

c log nNowwe claim that if A then A forq p

p q

c log n

R

q

Indeed log A log A c log ndr log loglog So by

q p

p

increasing p by at most log c log n we reached q with A and bythe

q

same token another increase of at most log c log n will giveus A

q

as required

The case X f g of the BKKKL theorem was proved by Kahn Kalai

and Linial in resp onse to a conjecture of BenOr and Linial Some words

on the pro of of this theorem are in place The pro of uses harmonic analysis on

EHUD FRIEDGUT AND GIL KALAI

n n

Z For a b o olean function f on f g consider the WalshFourier expansion

P

jS T j n

f f S u where u T Here we identify vectors in f g

S S

S n

with subsets of n in the standard way It can b e shown that the sum of inuences

P

f S jS j In order to show that A is large one has A is equal to

S n

f is concentrated in to show that a large p ortion of the WalshFourier transform of

high frequencies This is shown if all inuences are not to o large by applying a

certain hyp ercontractive estimate of Beckner In the next section we will show

that if all inuences are smaller than then Aisatleastc log

p

The pro of of the BKKKL theorem is similar and is based on a FourierWalsh

interpretation for inuences in arbitrary pro duct spaces Talagrand see remark

in Section found another pro of for the sp ecial case X f g He replaced the

p

Walsh functions u by an appropriate orthonormal basis for V p All these pro ofs

S n

give constants which are quite realistic

Some extensions

Consider now the situation when p itself is a function of nWe will describ e now

sharp forms of the ab ove theorems which shows that if A and A

p q

log p

c

If p n for some cwe cannot then q p o always hold when

log n

exp ect in general that q p opanditisaninteresting problem to understand

for which As this relation holds We need an improvement of the BKKKL Theorem

for the sp ecial case of X V p

n

Theorem For every function f V p f g with Prf t

n

there is a variable k so that

t log n

I k c

f

n p log p

This gives as ab ove a sharp up to constants form of Theorem

Theorem For every symmetric monotone property Aif A then A

p q

for q p c log p log p log n

Corollary Let P be any monotone property of graphs Then if P then

p

P for q p c log p log p log n

q

Pro of of Theorem We rely on the pro of from Note that the pro of of

Lemma in giv es for a monotone function on f g thatw f cp log p

p

To see this note that for X f g wehave the additional inequalities I j p

p f

mj

for every j in addition to the inequalities I j which always hold So

f

P

m

I j p log pp Substituting the improved upp er b ound for w f

f

j

I k and wereacha in relation of we get that kW k c p log p

f k

contradiction in the same way as in the original pro of by assuming that I k

f

tlog n

c for c suciently small

plogpn

SHARP THRESHOLDS FOR GRAPH PROPERTIES

The pro of of the BKKKL Theorem can b e mo died to give the following

n

If Theorem For every function f X f g with Prf t

I k for every k then

f

n

X

I k c t log

f

k

Pro of Again we rely heavily on the pro of of BKKKL Theorem and recycle the

constants c c Let I k Then as in the original pro of wehavethat

k f

more than half of the weightof kf k is concentrated where

n

X

jS j jS j jS j c t

n k

k

Substituting relation from in relation we can replace the right hand

P

n

It follows that more than half the weightof side of relation by c

k

k

kf k is concentrated where

n

X

jS jjS jjS j

n

 

jS j jS j c t

n k

k

This implies that

n n

X X

c t log t

k k

k k

P

n

Now assume that for every k and that c t log Using

k k

k

P

n

the convexity of the function x we get that the maximum value of

k

k

is attained when for k r and forkr where r

k k

P

n

c log c t log Therefore log t

k

k

n

Corollary L et A be a monotone subset of f g such that I k for every

A

p Then if A then A for q p c log log

p q

It follows from Theorem that if wehave a monotone subset A whichisinvariant

under a p ermutation group with the prop ertythatevery transitivity class of has

at least a elements then the assertion of Theorem holds with log n replaced

by log a Therefore the assertion of Theorem holds with a dierent absolute

constant for symmetric prop erties of random subgraphs of arbitrary vertextransitive

graphs in particular Cayley graphs on n vertices

The pro of of Theorem extends without change to

X X f gwith Prf Theorem For every function f X

n

if I k for every k then t

f

n

X

I k c t log

f

k

EHUD FRIEDGUT AND GIL KALAI

Q

n

X Note that for a monotone Assume nowthatX f g and put X

i k p

i

k

subset A of X A p I k A is a linear function of p with slop e

k A k

I k this is p erhaps the shortest derivation of the MargulisRusso Lemma It

A

follows by a similar argument to the pro of of Theorem that for every subset A

of V and every ntuple of probabilities p p p if A thenthe

n n p p p

n

 

following statement holds There is a vector of probabilities q q q suchthat

n

A and for every

q q q

n

 

jfi q p c log log gj

i i

Tightness of the results

BenOr and Linial constructed a trib es example to showthattheO log nn

lower b ound on inuences is sharp see also The following more general examples

show thatuptomultiplicativeconstants Theorem is sharp when p do es not

dep end on n and Theorems and are sharp even when p do es dep end on n

Let n m r Partition ninto sets T T T of size m and let A b e the set of

r

subsets of n whichcontain T for some iFor a given probability q take

i

log n log log n log log q

m

log q

Then A is close to eFor p q o let A t and then

q p

d A

p

t log t log np log p

dp

Therefore the interval q p where A is of length log q log q log n

p

There are many other examples for which Theorem is sharp For example

consider the prop erty of subsets S of nS contains an interval of length k k n

of consecutiveintegers mo dulo n Other examples are the prop erties of subgraphs

H of the graph of the ndimensional cub e H contains the graph of a k dimensional

face and as mentioned b elow H is connected

Graph prop erties Let P b e the prop ertyG contains a clique with k nvertices

For xed n and k let X b e the number of k cliques in Gn p The exp ected value

np

of X is

np

k

n



p n k p

k

Consider now an arbitrary function k nO log nandletq q nbesuch that

n k nqn Note that log q log n ik In this case Corollary

asserts that the length of the threshold interval is oq For p q o

X can b e approximated byaPoisson random variable with mean n k p and

np

nk p

ProbX e o This can b e justied along the lines of np

SHARP THRESHOLDS FOR GRAPH PROPERTIES

pp This shows that the actual threshold interval is of length prop ortional

 

logq n logq n

rather than q n to q n

log n log n

Bounded depth circuits There is an interesting connection b etween the com

plexity of Bo olean functions and their threshold b ehavior It follows eg from the

Hastad Switching Lemma see that Bo olean functions f that can b e ex

pressed by b oundeddepth p olynomial size circuits have large threshold intervals

Put ap f If f is expressed by a depthtwo circuit of size N then

p

limap ap s log N

s

In other words the length of the threshold interval is at least c p log N where

ap This result is tight Most of the examples describ ed in this section and

in the next section can b e expressed by depthtwo circuits of small size For circuits

d

of depth d the term log N should b e replaced bylog N This connection was

y Jo el Sp encer p ointed out to us byNogaAlonandb

The dependence of the threshold interval on the permutation

group

The content of this section is the fruit of collab oration with Aner Shalev

Problem For apermutation group on n how large can the threshold interval

be for monotone families A which are invariant under

Here by the threshold interval we mean the length of the interval p q where

A and A for a xed Given a p ermutation

p q

group S consider the following class of examples For every s nd a set S

n

jS j s such that the orbit of S under is minimal Consider the family A to b e

S

those subsets of nwhichcontain a set in the orbit of S We conjecture that up

to multipli cative constants such an example will give the largest threshold intervals

among monotone families invariant under

p

If S then the length of the threshold interval is prop ortional to nFor a set

n

X

be the set of r subsets of X Aswe already mentioned for graph prop erties X let

r

m

ie when S acting on the threshold interval can b e as large as c log n

m

m

n and we conjecture that it cannot b e larger For symmetric prop erties of r

m

uniform hyp ergraphs ie when S acting on the threshold intervals can

m

r

r

m

r 

b e as large as log nn and again we conjecture that it cannot b e larger

r

Another example of interest is the group PSLd q actingasapermutation

group on ddimensional pro jective space over a eld of q elements If q is xed

say q then considering the monotone families of all sets containing a subspace

of dimension t logd log log d log log n log log log n we get an example

EHUD FRIEDGUT AND GIL KALAI

and we conjecture that no monotone with threshold interval of length

log n log log n

symmetric prop erty with larger threshold interval exists

By forming wreath pro ducts one can construct groups with intermediate threshold

b ehavior b etween log n and n However we conjecture that for primitive

p ermutation groups there are some gaps in the p ossible b ehavior of the largest

threshold intervals for prop erties which are invariant under

The length of this interval is prop ortional to n for S and A but at least

n n

log n for any other primitive p ermutation group

The length T of the largest threshold interval satises c log n T

c log n for f g or for which tends to one as

a function of n in an arbitrary way

If do es not involve as factors large alternating groups then the length of

the largest threshold interval is prop ortional to log n w n where w n

log log n

This description follows from the following We conjecture that the value of the

largest threshold interval for invariant prop erties where is a primitive p ermutation

subgroup of S other than A or S is prop ortional to where

n n n

jT j

minfjT j jT j g

The family A for a set T which realize the minimum in the denition of seems

T

to have threshold interval of length prop ortional to jT j and this can b e proved

log log n

when jjn using the detailed knowledge of suchpermutation groups

As for upp er b ounds it seems that the issue is to show that if a Bo olean function f

has FourierWalsh co ecients which are smeared then the sum of inuences must

b e large The following or similar inequality is needed

n

Conjecture For every Bo olean function f f g f g

X X

f S log jf S j c f S jS j

Some connections to earlier work

Connectivity and the MargulisTalagrand isop erimetric formula Perhaps

the most extensively studied problems in random graph theory as well as in p er

colation theory on critical probabilities and threshold b ehavior are on connectivity

Erdos and Renyi found the critical probability p pnlognn for a random

graph in Gn p to b e connected and proved that the threshold interval is of length

op For connectivity of random subgraphs of the graph of the ndimensional cub e

Burtin found the critical probability p and results of Erdos Sp encer and

Bollobas show that the threshold interval is of length O n See pp

Margulis see also pp proved a sharp threshold prop ertyfor

connectivity of random subgraphs of k connected graphs as k and Talagrand

proved sharp er forms of Margulis result Neither the results of this pap er nor

SHARP THRESHOLDS FOR GRAPH PROPERTIES

the MargulisTalagrand theorem include the ErdosRenyi result as a sp ecial case

For random subgraphs of the the graph of the ndimensional cub e the Margulis

Talagrand theorem gives that the length of the threshold interval is O n and

Theorem gives O n which is sharp

The main to ol used by Margulis and sharp ened byTalagrand is the following Let

A fv A hv g be the vertexb oundary of A Margulis proved that

v

A A g p A

p v p p

for some p ositive function g p A in Itwould b e interesting to understand

p

the connection b etween the inuences of the variables and the quantity A

p v

A

p

Russos approximate law Corollary is a sharp version of a theorem of

Russo A weaker version of Theorem can b e derived also from Russos theorem

itself The derivation is not immediate since in order to apply Russos theorem it is

necessary to show that I k o for every k and every p which is not very close

f

to or This follows from

n

Lemma Let p p bexed LetBbe the Hamming bal l in f g con

P

n

taining al l the sets S such that jS j np Then for every monotone A I k

A

k

P

n

k I

B

k

Pro of Write S R for R S and jS j jRj

n

X X X X

jX j njX j k nk

I k p q S R p q S k np

A A A A

S R k

S jS jk

To maximize this sum A should include precisely those sets S with jS j np It can

b e proved b y a similar argumentthatifB is any Hamming ball around

P P

and A is any monotone set such that A B then I k I k

p p A B

q

P

n

It follows from Lemma that I k c and therefore if A is symmetric

A

pp

and p is not close to or to then all inuences are o

Remark After this pap er was submitted we learned ab out Talagrands pap er

which has some overlaps with the present pap er as well as with Talagrands

main result is a sharp form of BKKKLs theorem for the sp ecial case of V p He

n

also related this result via the MargulisRusso lemma to threshold phenomena and

to Russos approximate law His results include Theorem and the sp ecial case

of V p of Theorem

n

Talagrand shows that for the inuences of a Bo olean function f with pr obf

say

X

I k log I k C

f f

for some universal constant C This is remarkable b ecause for some other universal

P

constant D if a log a D then there exists a Bo olean function f with

i i

EHUD FRIEDGUT AND GIL KALAI

pr obf suchthatI k a for every i i n To construct f

f i

consider a trib es example with trib es of dierent sizes

KruskalKatona formulation A simplicial complex K is a collection of subsets of

n such that S K and R S implies that R K Letf K denotes the number

k

of sets in K of cardinality k Switching from the mo del where eachvertex is chosen

with probability p to the mo del where each set of cardinalitypnischosen with equal

probabilitywe obtain the following strong form of the KruskalKatona theorem for

simplicial complexes with transitive group action on the vertices A similar extension

of Macaulays theorem on complexes of monomials multisets also follows

vertices which is invariant under Theorem Let K be a simplicial complex with n

n

atransitive permutation group on its vertices If f K then f K

k r

k

n

for r k c log nn

r

Acknowledgement Wewould like to thank Nati Linial for suggesting the problem

and for many other useful suggestions Noga Alon JeKahn and Aner Shalev for

fruitful discussions and Itai Benyamini for p ointing out the pap er byTalagrand

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SHARP THRESHOLDS FOR GRAPH PROPERTIES

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Institute of Mathematics The Hebrew University of Jerusalem Givat Ram Jerusalem

Israel

Email address ehudfmathhujiacil

Institute of Mathematics The Hebrew University of Jerusalem Givat Ram Jerusalem

Israel and Institute for Advanced Studies Princeton NJ

Email address kalaimathhujiacil