Graph Generated Unionclosed Families of Sets

Emanuel Knill

Los Alamos National Lab oratory

Los Alamos NM

knilllanlgov

revised June

Abstract

S

e j E Let G b e a graph with vertices V and edges E Let F f

eE

E g b e the unionclosed family of sets generated by E Then F is the family

of subsets of V without isolated p oints Theorem There is an edge e E

1

such that fU F j U eg F This is equivalent to the following

2

assertion If H is a unionclosed family generated by a family of sets of

maximum degree two then there is an x such that fU H j x U g

1

H This is a sp ecial case of the unionclosed sets conjecture To put this

2

result in p ersp ective a brief overview of research on the unionclosed sets

conjecture is given A pro of of a strong version of the theorem on graph

generated families of sets is presented This pro of dep ends on an analysis

of the lo cal prop erties of F and an application of Kleitmans lemma Much

of the pro of applies to arbitrary unionclosed families and can b e used to

obtain b ounds on fU F j U eg F

Preliminaries

Unless stated otherwise all sets are assumed to b e nite Let N b e the set of

nonnegative integers Dene n fi N j i ng and m n fi

N j m i ng Let X b e a set A family of sets on X is a subset F of

X

the p ower set of X F is unionclosed intersectionclosed i for every

U V F U V F U V F The family F is a p oset where the memb ers

of F are ordered by inclusion For Y X let

F fU F j U Y g

Y

b e the family induced by F in Y let

F fU F j U Y g

Y

b e the family induced by F above Y let

F fU Y j U F g

Y

b e the restriction of F to Y and let

F fU n Y j U F g

nY

b e the restriction of F away from Y If Y fxg is a singleton the set brackets

are omitted The degree of Y in F is dened as d Y F For families

F Y

of sets F and G let F G fU V j U F V G g and F G fU V j U

F V G g The transpose or dual of F is the collection of sets F on F dened

by

hF j x X i F

x

where the F are counted with multiplicities A collection of sets is called

x

simple if each memb er o ccurs only once ie if it is a family of sets If the

dual of F is not simple then there are elements of X which are not separated

by any memb er of F Call F primitive on X if the dual of F is simple and

S

F X

Let P b e a p oset The dual P of P is P with the reverse ordering A

map f from P to a p oset Q is orderpreserving i x y implies f x f y

P P

The set of orderpreserving maps from P to Q is denoted by Q Q is a p oset

with the p ointwise ordering i for all x x x The element x P

covers y P i x y and x z y implies z x or z y For A P let

A fx P j x y for some y Ag

P

A fx P j x y for some y Ag

P

Thus A is the order ideal and A is the order lter generated by A Sub

P P

scripts are omitted if the p oset is clear from context To simplify notation for

x P let x fxg and x fxg

If P is closed under least upp er b ounds then P is a joinsemilattice

with the least upp er b ound of x and y denoted by x y the join of x and y

Similarly if P is closed under greatest lower b ounds then P is a meetsemilattice

with the greatest lower b ound of x and y denoted by x y the meet of x and

y A lattice is b oth a joinsemilattice and a meetsemilattice Recall that any

nite joinsemilattice can b e made into a lattice by adding a least element if

necessary and similarly for meetsemilattices The meet join of elements in

a joinsemilattice meetsemilattice is understo o d to b e the meet join in this

extension

If L is a lattice and x L satises that u v x implies u x or v x

then x is joinirreducible Every element of L can b e represented as the join

of the joinirreducible elements b elow it The set of joinirreducible elements

excluding the least element is denoted by J L Meetirreducible elements are

dened dually M L denotes the set of meetirreducible elements of L excluding

the largest element

There are canonical corresp ondences b etween nite semilattices and

primitive union or intersectionclosed families of sets Clearly every union or

intersectionclosed family considered as a p oset is a semilattice Let L b e a

meetsemilattice Let

F L fx J L j x Lg

L

Then F is a primitive intersectionclosed family of sets If L is a joinsemilattice

then the corresp onding unionclosed family of sets consists of the family of

complements in M L of memb ers of F L

S

Let F b e a primitive unionclosed family of sets on X F The

S

meetirreducible elements of F are given by the sets M F The

x

X nfxg

union generators of F are given by GF J F fg This is a family of sets

with the prop erty that no memb er is the union of any collection of the other

memb ers If GF G dene F G F

A family of sets G is a graph i for every U G U The

memb ers of G are referred to as edges The graph G is simple i for every U G

U A graph generated unionclosed family of sets satises that GF is a

graph

Unionclosed families of sets

The following problem has b een attributed to PFrankl Duus Stanley

Problem The unionclosed sets conjecture Let F be a unionclosed

family of sets with at least one nonempty set Does there always exist an ele

F ment x such that d x

F

The family of sets consisting of only the empty set do es not satisfy the

assertion of this problem Henceforth all families of sets and semilattices are

assumed to have at least two memb ers It is straightforward to check that the

n

unionclosed conjecture holds for F if n

Using duality and the corresp ondence b etween unionclosed families of

sets and joinsemilattices one arrives at other questions equivalent to the union

closed sets conjecture The rst such question is obtained by complementing

the sets of a unionclosed family This yields an intersectionclosed family order

isomorphic to the p oset dual

Problem Let F be an intersectionclosed family of sets Does there always

S

F exist an element x F such that d x

F

The second equivalent question is obtained from the rst by using the cor

resp ondence b etween intersectionclosed families and meetsemilattices and by

observing that if F is a primitive intersectionclosed family then the elements

S

of F are in onetoone corresp ondence with J F

Problem Let L be a meetsemilattice Does there always exist a member

x J L such that x L

There is a corresp onding version for joinsemilattices The last equivalent ques

tion to b e given here is obtained by considering a unionclosed family of sets

with as a meetsemilattice

Problem Let F be a unionclosed family of sets with F Is there

always a member U of GF such that F F

U

The unionclosed sets conjecture can b e generalized as follows Knill

Let P b e a p oset and let p b e the numb er of lters of P A meetsemilattice L has

P P

x L p the P density property i there exists x J L such that

The quantity on the left of this inequality is called the P density of x in L

The density of x is the density of x The density property is the density

prop erty

Problem The P density problem What meet semilattices have the P

density property

For P this reduces to Problem Ie L has the density prop erty i it

satises the assertion of Problem If L has the density prop erty with witness

a J L then we say that L a has the density prop erty Again the trivial

example the oneelement semilattice do es not have the P density prop erty

Some results related to the unionclosed sets conjecture are given next

Recall that any intersectionclosed family is a meetsemilattice

The next two results have b een rediscovered a few times and are given

in

S

Theorem If the intersectionclosed family F contains F n fx y g for

S

some x y F then F has the density property

P

Let S F U

U F

F of the members of the intersection Theorem If the average size S F

S

closed family F is at most then F has the density property F

Theorems and are obtained by double counting They can b e

generalized substantially a long overdue analysis of the technique is given by

Po onen These ideas can b e used to establish lower b ounds on counter

examples to the unionclosed sets conjecture currently these b ounds are ab out

S

F F and

The motivation underlying the work presented in Section is related to

the ideas in One of the main results of Theorem characterizes union

closed families F which satisfy that for every unionclosed extension G F there

S

G Here we consider extensions G of F in exists an x F with d x

G

the context of Problem In terms of the unionclosed sets conjecture this

can b e seen to involve mo difying F by adding new generators and adding new

elements to the generators of F

S

For a given unionclosed family F with F X dene

S F

sF X

F X

S

then F satises the unionclosed sets As in Theorem if sF F

X conjecture Wojcik studies the prop erties of s min fsF X j

m

S

log m

2

m F X g He conjectures that s o as m In fact

m

m

this is one of very few so far unproven apparently proper consequences of the

unionclosed sets conjecture Wojcik also gives an exact version of his conjecture

which follows from a slight extension of the unionclosed sets conjecture

Wojciks conjecture can b e strengthened as follows unpublished For

P

ln

i

each n let n bn b e the binary expansion of n where n and

i

i

l n or bn Let

ln

U n fi j i l n and bn or i l n and bn g

i i

The function U is a bijection from N to nite subsets of N It induces an

ordering on such sets which is closely related to the lexicographic orderings

If k l and U i U k U l then i l It follows that the families

U n fU U U n g are unionclosed Let

t minfS F j F ng

n

Problem Is it true that t S U n

n

This can also b e shown to b e a consequence of the unionclosed sets

conjecture It seems p ossible that the families U n minimize max d x for

x F

F n This would imply the unionclosed sets unionclosed families F with

n

conjecture Note that F is the family of all subsets of n Problem

is reminiscent of the KruskalKatona theorem according to which the sizes of

the shadow and the ideal generated by a family of n k sets are minimized by

the family which consists of the rst n k sets in the squashed ordering see

ch

Many of the standard classes of lattices can b e shown to satisfy P

density prop erties A lowersemimo dular coatom of the lattice L is a coatom a

with the prop erty that for every w L with w a w covers a w

Theorem The fol lowing classes of lattices have the indicated density prop

erty

Distributive lattices P density for al l P

Modular lattices P density for al l P

Geometric lattices ndensity for al l n this is stated for n in

see

Lattices where every x is complemented density property

Lattices with a lowersemimodular coatom P density for al l P

Lattices of height h with h J L P density for al l P

Selfdual lattices density property

Note that the third class includes the second which includes the rst

and the fth class includes the rst two The result for selfdual lattices follows

from the observation that for every lattice L either L or L has the density

prop erty The other results dep end on the existence of certain matching prop

P

erties in a lattice If F is a lter of P then let TL F a f L j x

a i x F g The memb ers of TL F a are called the orderpreserving maps

of type F a We say that L has the P matching property i there is a join

irreducible a such that for every lter F P there is a decreasing onetoone

map TL P a TL F a A map is decreasing i x x L has the

ful l P matching property i there is a joinirreducible a such that for all lters

F G P there is a decreasing onetoone map TL G a TL F a

The P matching prop erties imply the P density prop erty There are nontrivial

lattices which do not satisfy any P matching prop erty One such example is

given by the unionclosed family generated by the edges of the p entagon

Here are some of the results ab out preservation of the density and

matching prop erties under lattice op erations obtained in

Theorem If the meetsemilattice L has the P density property then so

does L M for any meetsemilattice M

For p osets P and Q P Q is the disjoint union of P and Q ordered

by the union of the orders on P and Q

Theorem Preservation of matching properties for semilattices

If L has the ful l P matching property then so does L M

If L has the ful l P matching property for a J L and I is an ideal of

L with a I then I has the ful l matching property for a

Q

If L has the ful l P matching property then so does L

L has the ful l P Qmatching property for a J L i L has the ful l P

and the ful l Qmatching property for a

If L has the P and the Qmatching property for a J L then L has the

P Qmatching property for a The reverse implication holds if a is an

atom

Theorem can b e used to show that if the b ound of Theorem

can b e improved to o then the statement of the P density problem is

p

true In also suces to show that the P density prop erty is satised by atomic

lattices

Matching prop erties are also preserved by certain sub direct pro ducts

which preserve local prop erties of lattices This notion of lo cality can b e made

precise by using the representation of lattices by unionclosed families of sets

Let L b e a unionclosed family of sets The lattice neighborhood N U of

L

S

U J L is the unionclosed family generated by fg fA J L j A U g

Here is one of the results that can b e obtained

Theorem If x J L and N x is a geometric lattice then L has the

L

ndensity property with witness x

If the unionclosed sets conjecture is true then for every unionclosed

family F of sets there exists a cover Y of the nonempty memb ers of F with

at most dlog F e elements Recall that Y is a cover of G i Y intersects

every memb er of G Assuming the unionclosed sets conjecture Y is obtained

by rst chosing y such that y covers at least half of the memb ers of F Since

F F is unionclosed one can chose y such that y covers at least

F nfy g

1

half of the memb ers of F Continuing in this fashion the desired cover Y is

obtained The fact that such a set Y exists can b e shown without making use

of the unionclosed sets conjecture It suces to observe that for every minimal

cover Y of the nonempty memb ers of F F fg is Bo olean This implies that

Y

Y log F A related result gives an upp er b ound on the minimum

P density of joinirreducibl es in a semilattice

Theorem The minimum P density of a joinirreducible in the meet

semilattice L satises asymptotical ly as L goes to innity

log L

p

An indication that the b ound of Theorem is weak is given by

the fact that it is asymptotically b est p ossible for the generalization of meet

semilattices to multimeetsemilattices where every memb er other than is

assigned a multiplicity

Before intro ducing the main result of the pap er here is another inter

esting result

Theorem For every xed lattice L there is an N such that for al l n N

L has the ndensity property

This result is proved by considering the Zeta p olynomial after elimi

nating the obvious cases and lattices of height J L

Graph generated unionclosed families of sets

The main result proven in this pap er concerns unionclosed families generated

by a graph and Problem The most easily stated version of the result is as

follows

Theorem If F is a graphgenerated unionclosed family of sets with F

F F then there is a set U GF such that

U

The pro of of Theorem given here actually shows a substantially

stronger result of a lo cal nature Instead of assuming that GF is a graph

it suces that there is a U GF such that an appropriate neighb orho o d of

U is graphgenerated and U has lo cally minimal degree The pro of involves

estimating F F from lo cal prop erties of F These estimates are very

U

general and can b e applied to arbitrary unionclosed families

Denition Let G b e a graph and U G The degree in G of U is

fV G n fU g j V U g d U

G

Denition Let F b e a unionclosed family of sets The closure in F of the

set X is given by

S S

X F fV GF j V X g

F X

The set of isolated elements of X is

X X n X

F F

In terms of hyp ergraphs X is the set of isolated p oints of the hyp ergraph

F

H F F X For any subfamily H of F the density of H in F is the ratio

X

Observation Let F be a unionclosed family of sets The fol lowing are

equivalent

i U F

ii U U

F

U iii

F

S

Let F b e a unionclosed family of sets such that F and let U F

Recall that N U the lattice neighb orho o d in F of U is the unionclosed

F

family generated by the empty set and the memb ers V of GF with V U

S

The third lattice neighb orho o d is given by N U N N U The

F F

F

S

second one which consists of the sets of F included in N will not b e used

F U

here

We now develop the technique for estimating the density of F in F

U

This estimate dep ends only on the third lattice neighb orho o d N U

F

Denition Let H b e a family of sets The density in H of the set X is

H

X

X

H

H

Thus X is the density of H in H The recipro cal of X is denoted

H X H

by X Let U

H F

Denition The unionclosed family F is a conservative extension of F U

S

i there is a nonempty unionclosed family of sets H such that H U

and F F H

Asso ciativity of for families of sets yields

Observation The extension relation is transitive ie if F is an extension

of F U and F is an extension of F U then F is an extension of F U

Since F F fg

Observation F is an extension of F U

Denition Let

U j F is an extension of F U g inf f

F

Observation

Observation If U J F and then F U has the density prop

erty

The goal is to nd lower b ounds on in terms of the lo cal prop erties

of F at U To this end let F b e an arbitrary extension of F U and consider

F

U

F

F

U

Denition For X U let

X fY U j X Y F g T

F

We have

X

X F T

F

X F

nU

and

F F

U

nU

where equality holds if U F since F is closed under union with memb ers of

F This gives

P

X T

F

X F

nU

U

F

F

nU

U

We will determine Note that if U F then this is an identity so that

X a lower b ound for T which is indep endent of F

F

Lemma Let X F If Y U satises

nU

i X Y U Y

F

ii X Y X U n U

F F

then X Y F

Conditions i and ii are indep endent of F Observe that i is

equivalent to

X Y U

F

and ii is equivalent to

X Y n U X U

F F

Lemma b elow gives conditions equivalent to i and ii in terms of the

family of generators of F

Pro of Since X F there exists Z U such that X Z F Since F is

nU

an extension of F U there is a unionclosed nonempty family H such that

S

H U and F F H We have X Z A B for some A F and

B H We can assume that A X Z Let A X Y By i

F F

A U Y If we can show that X n A B then X Y A B F and

we are done

Let x X n A Then x X Y n U By ii x X U n U

F F

Since Z U X U X Z which yields x X Z n U Since

F F F

X Z n U X n A B we have x B as desired

F

Denition For every set X with X U let E X consist of the

F U

subsets Y of U satisfying the conditions of Lemma

E X fY U j X Y U Y and

F U F

X Y X U n U g

F F

Let E X E X

F U

Example Supp ose that F is the unionclosed family of sets generated by

the edges of the graph depicted in Figure and the empty set

x

t t

x

S S

S S

S S

t t t S S

x

a b S

U

S

S

t t S

x x

Figure

Then

n o

J F fa bg fx ag fx ag fx ag fx bg fx bg fx x g

Let U fa bg We have

n o

E fx x g fbg fa bg

n o

E fx g fag fbg fa bg

n o

E fx x g fa bg

X for every X F Using inequality By Lemma E X T

F

nU

we get

P

E X

X F

nU

U

F

F

nU

Denition Let

P

E X

X F

nU

U

F F

F

nU

U Let

F F F

By inequality

U Observation

F F

We can dene E X in terms of the set of generators of F

Lemma The set Y is in E X i

i for every x Y there exists V J F with x V X Y

ii for every V J F if V n U X then for every x V n U there

exists V J F with x V X Y

Pro of In fact condition i of Lemma is equivalent to i and condition

ii of Lemma is equivalent to ii

If x W F then there is a generator V J F such that x V

W This implies that X Y U Y i i holds hence i i i

F

Supp ose that X and Y satisfy ii Let V J F and V n U X

Then V X U By ii X Y X U n U Hence if x V n U

F F F

then x X Y which implies that there is a generator V of F such that

F

x V X Y Thus ii holds

Conversely supp ose that X and Y satisfy ii We show that X

F

Y X U n U Let x X U n U Then there exists V J F such

F F

that x V X U We have V n U X so by ii there exists V J F

with x V X Y This implies that x X Y as desired

F

Denition The neighborhood in F of a set X is given by

S

N X X fV J F j V X g

F

Let N N U and N N N U Observe that if U F then N

F F F

S S

U N U and N N

F

F

Lemma The family F is an extension of N U U

F

Pro of Let H b e the unionclosed family of sets generated by the empty set

and the generators V of F with V N U Then F N U H and

F F

S

H U This expresses F as an internal sub direct pro duct of N U

F

and H

Theorem Let F be an extension of F U Then

U U

3

F F

N U F

F

Pro of By Lemma and Observation F is an extension of

N U is welldened By Lemma and by de U U so that

3

U F N

F

F

nition of N U whether a given subset of U is in E X dep ends only on the

F

generators V J F with V N U Since J N U J F N U

F F F

Lemma implies that

E X E 3 X

F U

N U U

F

for every X disjoint from U The result now follows by denition of U

S

Denition Let D F An extension F of F U minimizes in D i

S S

F D and for every extension F of F U with F D

F F

S

Theorem Let D F There exists an extension F F H of F U

such that

D nU

i H is a lter of

ii F H

nU

iii F minimizes in D

D nU

Pro of Note that i implies ii Let H b e a lter of Since F

F H H For the reverse inclusion let V F H Then there is a

nU

W H with W V n U hence V n U H

S

Let F G b e an extension of F U such that G D n U and

F G minimizes in D Since F F G F G we have

nU nU nU

This implies that we can assume G F G

F G

F F G nU

nU

D nU

Let H b e the lter of generated by G and let F F H We

which implies that F is as desired show that

F G

F

For X G let

P X fY H j Y X g

G

Lemma Let X Y G with X Y Then P X P Y

Pro of Dene for Z P Y

Z Z n Y X

Since every Z P Y includes Y is a onetoone map To show that maps

P Y into P X let Z P Y Let X Z and supp ose that X X

G

Then X X By denition of Z X Y Let Y Y X Then Y G

and Z Y Y contradicting Z P Y

Lemma Let X G If Y P X then E Y E X

Pro of We show that for every Z U X Z Y Z Let Z U

F F

Let X X Z and Y Y Z The inclusion X Y implies

F F

X Y Since Y F and X G we have Y X F G so that

Y X n U Y n U X G Since Y P X and Y Y n U X it

follows that Y n U X Y X Using Y U Z we get Y X Z

G

hence Y X

The identities X Z Y Z and X U Y U

F F F F

imply that Z E X i Z E Y as required

For n let

G fX G j P X ng

n

By Lemma the G are lters of G Since F G G this implies that

n

nU

if G then F G is an extension of F U such that F G G

n n n n

nU

Let N b e the maximum value of P X Then G G We use Lemmas

N

and and the fact that the family fP X j X G g is a partition of H to

compute

F

P

E X

X F

nU

F

F

nU

P

E X

X H

P

X H

P

P X E X

X G

P

P X

X G

P P

N

E X

X G nG

n

n

P P

N

X G nG

n

n

P P P

N

E X E X

X G X G

n

n

P

N

G G

n

n

P

E X G and if G Dene We have

n F G F G

F

X G

P

then E X G and This yields

F G n F G F G

X G

n n

n

P

N

G G

F G F G n

n n

F

P

N

G G

n

n

P

N

G G

F G n F G

n

P

N

G G

n

n

P

N

G G

n

n

F G

P

N

G G

n

n

F G

which implies that F satises iii as required

Lemma Let X Y be sets disjoint from U If Y N X N then

E X E Y

Pro of By Lemma whether a given subset of U is in E Z dep ends only

on the generators of F included in N This implies that E Z dep ends only

on Z N

We strengthen Theorem

S

Theorem Let D F There is an extension F F H of F U

such that

D nU

i H is a lter of

ii F H

nU

iii F minimizes in D

iv F fD n N g

2

nN

Pro of By Theorem there is an extension F G of F which satises i

ii and iii Let H G fD n N g We show that F F H is as required

By construction F satises i ii and iv For X H let

P X fY G j Y N X N g

Then fP X j X Hg is a partition of G If Y P X then by Lemma

D nU

E Y E Y N E X N E X Since G is a lter of if X Y

then P X 2 P Y 2 Since P X 2 fX N g for each X H this

nN nN N

P X P Y Let implies that if X Y then

H fX H j PX ng

n

Then H is a lter included in H for each n and H H Let N b e the

n

maximum value of P X By assumption on F G Supp ose

F H F G

n

that for some n Then

F H F G

n

P

E X

X G

F G

G

P

P X E X

X H

P

P X

X H

P P

N

E X

n X H

n

P P

N

n X H

n

P

N

H

n F H

n

n

P

N

H

n

n

P

N

H

n F G

n

P

N

H

n

n

P

N

H

n

n

P

F G

N

H

n

n

F G

which is imp ossible Hence for each n F H also minimizes in D In

n

particular F minimizes in D as required

Corollary There is an extension F N U H of N U U such

F F

that

2

N nU

i H is a lter of

ii F H

nU

U 3 iii if F is an extension of F U then U

F F

N U F

F

S

Pro of Let D F and let F G b e an extension of F U satisfying the

conditions given in Theorem By Theorem F G is an extension of

N U U and

F

U 3 U

F F G

N U F G

F

By condition iv of Theorem F G 2 is isomorphic to F G This

N

and Lemma imply that

3 U 3 U

N U F G N U F G

2

N

F F

U G for some nonempty U U F N Since F is an extension of N

F F

S

with G U The family H G G is a lter of unionclosed family G

D nU

with H fD n N g Let H H Then

2 2

nN N

F G 2 N U G G 2

N N

F

N U H

F

so that F N U H is the required extension

F

Corollary implies that to determine the minimum p ossible value

U F can b e replaced by N U thus assume that F N U of

F F

F F

Henceforth we assume that U J F This implies that for every X

disjoint from U U E X so that E X To show that F U has

the density prop erty it suces to show that for every extension F of F U

F

Theorem The fol lowing are equivalent

i There exists an extension F of F U such that

F

2

N nU

ii There is a lter H of such that and for every minimal

F H

E X member X of H

Pro of Assertion ii implies i Supp ose that i holds By Corollary

2

N nU

there is an extension F H of F U such that H is a lter of and

2

N nU

Let H b e a minimal lter of such that Supp ose

F H F H

that there is a minimal memb er X of H such that E X Let H H n fX g

Note that the assumption on X and imply that H The family

F H

2

N nU

H is a lter of and

P

E Y

Y H

F H

H

P

E Y E X

Y H

H

P

E Y

Y H

H

F H

where we used the fact that if a b c and ac b then a bc

ac This contradicts the minimality assumption on H so that H is as desired

If there is no lter H satisfying the conditions in assertion ii of The

U for every extension F of F U so that F U has orem then

F

the density prop erty This yields

Observation If every extension F H of F U such that

2

N nU

i H is a lter of

E X ii for every minimal X H

satises then F U has the density property

F H

Let H b e an arbitrary lter satisfying i and ii of Observation

and let F F H So far the discussion did not require any additional

assumptions on J F We now assume that J F is a graph Thus for some

S

a b F U fa bg Assume that a b and let

N fx N n U j fx ag J F and fx bg J F g

a

N fx N n U j fx ag J F and fx bg J F g

b

N fx N n U j fx ag J F and fx bg J F g

ab

Then N n U N N N see Figure

a b ab

Lemma If X H then X N and X N

a b

Pro of Let X H Let X b e a minimal memb er of H such that X X

Supp ose X N Then either fbg E X if X N N or

a ab b

N

ab

t t p p p

b

B

B b

t t

b

Q

B

b

Q

b

B

p p

Q

b

Q

p t t p Q B b

N

N

b

a

Q

p p

a b

Q

Q

Q

t t Q

Figure

E X if X N N Since fa bg E X this contradicts

ab b

E X Thus X N By symmetry X N and we are done

a b

Corollary If X H and Y U then X Y U Y

F

Pro of Let X H and Y U Supp ose that a Y By Lemma there

exists x N X so that fx ag F and fx ag X Y which implies that

a

a X Y Similarly if b Y then b X Y as required

F F

Denition Let Y U and x N n U Let E Y x consist of the subsets X

of N n U such that there is an edge fx y g J F with y X Y or y x

Dene

E Y E Y x

xN nU

2

N nU

Observation For Y U and x N n U E Y x is a lter of

Observation If X E Y then X and Y satisfy condition ii of

Lemma

Lemma If X E Y H then Y E X

Pro of Let X E Y H By Observation X and Y satisfy ii of

Lemma By Corollary X and Y satisfy i of Lemma The result

follows by the pro of of Lemma

S

Denition Let U b e a lter of Dene

U

U

S

S

Thus U is the density of U in

Theorem

X Y

E Y x U

F F

Y U

xN nU

Pro of Let us start b e recalling Kleitmans lemma This result has many

applications Its pro of can b e found in Anderson

Theorem Kleitman Let X be an nset If F and F are lters in

X

then

F F F F

n n n

X

This says that the density in of the intersection of two lters is

at least the pro duct of the densities of each

Returning to the pro of of Theorem it follows from Kleitmans

lemma that for every Y U

H E Y H E Y

Lemma and the fact that U E X for every X H yield

X X

E X fX H j Y E X g

Y U X H

X

fX H j U E X g fX H j Y E X g

Y U

X

H H E Y

Y U

Using H F we get

nU

P

E X

X H

F

H

P

H H E Y

Y U

H

X

H E Y

H

Y U

X

E Y

Y U

Multiple applications of Kleitmans Lemma yield

Y

E Y E Y x

xN nU

and the result follows

Dene

J F fV J F j V g

For x N n U dene

Ax ffx y g j y U and fx y g J F g

Ax Let nx

Lemma Let x N n U Then

nx

i E x

nx

ii If x N then E fag x and E fbg x

a

nx

iii If x N then E fag x and E fbg x

b

iv If x N then E fag x E fbg x

ab

Pro of Consider E x If fxg J F then E x If fxg

J F then

2 2

N nU N nU Ax

n E x

We have

2 2

N nU Ax N nU Ax

which implies that

nx

E x

The remaining cases are proved similarly

Recall that d U is the degree in the graph G of U If U has minimal

G

degree in J F then F U has the density prop erty

Theorem If for every edge V J F with V U d V

J F

U d U then for every extension F of F U

F F

J F

Pro of We can assume that F F H where H is a lter satisfying i

and ii of Observation Let n d U n N n N and

a a b b

J F

n N Then

ab ab

n n n n

a b ab

Let x N We have d fa xg n and d fa xg n n nx

a a ab

J F J F

This gives nx n n n n n Similarly if x N then nx

a ab b ab b

n n Lemma and the assumptions on H imply that n and n

a ab a b

By Theorem and Lemma

Y Y

E fag x E fbg x

F

xN nU xN nU

Y Y

nx nx

xN xN

a

b

n n

a

b

n n

a

b

n n

a

b

n n

a

b

Let c and c It remains to show that

c c Without loss of generality assume that n n If n then

a b a

n n

b b

c c

m

Supp ose that n Observe that m for m Using the fact that

a

b

a ba for a and b Lemma we obtain

n n

b b

n c c

a

n n

b b

n

a

n n

a

b

n

a

as required

b

Lemma If a and b then a ba

Pro of The result is true for a Dierentiating b oth expressions relative

to a yields

d d

b b

a b a b ba

da da

The result follows

If J F then F is generated by oneelement sets so that F is a

Bo olean lattice Thus by Observation Theorem has b een proved In

fact we have the following stronger result

Theorem Let F be a unionclosed family of sets and U J F Let

G fV J F j V U g If

i U

ii G is a graph

iii there is a simple graph G J F such that for every V G with

U V d V V G and d

G G

then F U has the density property

Pro of Supp ose that N is a unionclosed family such that F N N U

F

Then F is an extension of N U see the pro of of Lemma

Lemma shows that E X E X for every X with X U

F U N U

Therefore

U U

N F F F

for every extension F of F U Let N b e the unionclosed family generated

by G G fg Then F N N U Theorems and show that

F

U for every extension F of F U It follows that F U has the

N F

density prop erty

References

I Anderson Combinatorics of Finite Sets Clarendon Press Oxford

D Daykin in I Rival ed Ordered Sets Proceedings of the NATO Ad

vanced Study Institute Nato Advanced Study Institutes Series Volume

D Reidel Publishing Company Dordtrecht

D Duus Matchings in mo dular lattices J Combin Theory A

D Duus in I Rival ed Graphs and Order D Reidel Publishing Com

pany Dordrecht

G Gratzer General Lattice Theory Birkhauser Verlag Basel

DJ Kleitman Families of nondisjoint subsets J Combin Theory

E Knill Generalized Degrees and Densities for Families of Sets PhD

Thesis University of Colorado at Boulder

JPS Kung Matchings and Radon transforms in lattices I Consistent

lattices Order

JPS Kung Radon transforms in combinatorics and lattice theory

Combinatorics and Ordered Sets Contemp orary Mathematics Volume

American Math So ciety Providence RI

B Po onen Unionclosed Families Journal of Combinatorial Theory Se

ries A

B Sands in I Rival ed Ordered Sets Proceedings of the NATO Ad

vanced Study Institute Nato Advanced Study Institutes Series Volume

D Reidel Publishing Company Dordtrecht

DG Sarvate and JC Renaud On the unionclosed sets conjecture Ars

Combinatoria

DG Sarvate and JC Renaud Improved b ounds for the unionclosed

sets conjecture Ars Combinatoria

R P Stanley Enumerative Combinatorics Vol I WadsworthBro oks

Cole Monterey Cal

P Winkler Unionclosed sets conjecture Australian Mathematical Soci

ety Gazette

P Wojcik Density of unionclosed families Discrete Mathematics