Minimal weight in union-closed families

0.15 MB·English

by  Victor Falgas--Ravry

Minimal weight in union-closed families
Victor Falgas–Ravry∗
January 14, 2011
1
1
0
2

n Abstract
a
J Let Ω be a finite set and let (Ω) be a set system on Ω. For
S ⊆ P
3 x Ω, we denote by dS(x) the number of members of containing x.
1 ∈ S
Along-standingconjectureofFrankl[6]statesthatif isunion-closed
1 S
] then there is some x Ω with dS(x) 2 .
O ∈ ≥ |S|
We consider a related question. Define the weight of a family
C S
to be w( ) := A. Suppose is union-closed. How small can
A∈S
. S | | S
h w( ) be? Reimer [11] showed
S P
t
a 1
m w( ) log2 ,
S ≥ 2|S| |S|
[

and that this inequality is tight. In this paper we show how Reimer’s
1
bound maybe improvedifwe havesomeadditionalinformationabout
v
9 the domain Ω of : if separates the points of its domain, then
S S
8
5 Ω
2 w( ) | | .
S ≥ 2
. (cid:18) (cid:19)
1
0 This is stronger than Reimer’s Theorem when Ω > log2 . In
1 |S| |S|
addition we construct a family of examples showing the combined
1 p
: bound on w( ) is tight except in the region Ω = Θ( log2 ),
v S | | |S| |S|
where it may be off by a multiplicative factor of 2.
i p
X Our proofalso gives a lower bound on the averagedegree: if is a
S
r point-separating union-closed family on Ω, then
a
1 1
dS(x) log2 +O(1),
Ω ≥ 2 |S| |S|
| |x∈Ω
X p
and this is best possible except for a multiplicative factor of 2.
∗SchoolofMathematicalSciences,QueenMary,UniversityofLondon,LondonE14NS,
England
1
1 Introduction
LetΩbeafiniteset. Wemay identifyX Ωwithitscharacteristic function
⊆
and consider a collection of subsets of Ω as a family of functions from Ω into
0,1 . For such a family (Ω), we refer to Ω = Ω(S) as the domain of
{ } S ⊆ P
. Note that the domain of a set system is not uniquely determined by
S S
knowledge . Therefore when we speak of ‘a set system ’, we shall in fact
S S
mean ‘a pair ( ,Ω), where (Ω)’ so that the domain of is implicitly
S S ⊆ P S
specified.
We also let V(S) := A be the set of all elements x Ω which
A∈S ∈
appear as a member of at least one set A . For x Ω we denote by
S ∈ S ∈
d (x) the number of members of containing x. We call d (x) the degree
S S
S
of x in .
S
A set system is union-closed if it is closed underpairwise unions. This
S
is essentially thesameas beingclosed underarbitrary unionsexcept thatwe
donotrequire tocontain theemptyset. In1979, Frankl[6]madeasimple-
S
sounding conjecture on the maximal degree in a union-closed family. This
remains open and has become known as the Union-closed sets conjecture:
Conjecture 1 (Union-closed sets conjecture). Let be a set system on
S
some finite set Ω. Then there is an element x Ω which is contained in at
∈
least half of the members of .
S
(An equivalent lattice-theoretic version also exists. See for example Abe
and Nakano, Poonen or Stanley [1, 10, 13].)
Very little progress has been made on Conjecture 1. A simple argument
due to Knill [7] establishes that for any union-closed family with = m,
S |S|
therealways existssomexcontained inatleast m membersof . W´ojcik
log2m S
[14]improvedthisbyamultiplicativeconstant. Theconjectureisalsoknown
to hold if < 40 (see [9, 12]) or V( ) < 11 (see [8, 2]), if > 5 2|V(S)|
|S| | S | |S| 8×
(see [3, 4, 5]), or if contains some very specific collections of small sets
S
(see [8, 2]).
In a different direction, Reimer [11] found a beautiful shifting argument
to obtain a sharp lower bound on the average set size of as a function of
S
. We state his result here.
|S|
Theorem (Reimer’s Average Set Size Theorem). Let be a union-closed
S
family. Then
1 log
A 2|S|
| | ≥ 2
|S| A∈S
X
with equality if and only if is a powerset.
S
2
Define the weight of a family to be
S
w( ) := A
S | |
A∈S
X
= d (x).
S
x∈Ω
X
We shall think of Reimer’s Theorem as a lower bound for the smallest
possible weight of a union-closed family of a given size. Let be a union-
S
closed family. In this form, Reimer’s Theorem states that
log
w( ) |S| 2|S|
S ≥ 2
with equality if and only if is a powerset. The purpose of this paper is to
S
show how we may improve this inequality if we have some additional infor-
mation about Ω( ). As a corollary, we also give asymptotically tight (up to
S
a constant) lower bounds on the average degree over Ω, 1 d (x).
|Ω| x∈Ω S
As we remarked earlier, Ω( ) is not uniquely specified by . For ex-
S P S
ample, Ω( ) could contain many elements which do not appear in . This
S S
would bring the average degree in Ω arbitrarily close to 0. Restricting our
attention to V( ) does not entirely resolve this problem: pick x V( ).
S ∈ S
Replacing every instance of x in a member of by a set x ,x ,...x for
1 2 M
S
some arbitrarily large M gives us a new union-closed family ′ with the
S
same structure as but with average degree over V( ′) arbitrarily close to
S S
d (x).
S
Thustosayanythinginterestingaboutaveragedegree,weneedtoimpose
a restriction on and its domain. In particular we want to make sure that
S
no element of Ω( ) is ‘cloned’ many times over. We make therefore the
S
following natural definition.
Definition. A family separates a pair (i,j) of elements of Ω( ) if there
S S
exists A such that A contains exactly one of i and j. is separating if
∈ S S
it separates every pair of distinct elements of Ω( ). If Ω(S) = n and is
S | | S
separating, we say that is n-separating.
S
Recalling our identification of sets with their characteristic functions,
S
is separating if and only if it separates the points of Ω( ) as a family of
S
functions Ω 0,1 .
→ { }
Trivially, a family of size = m can be at most 2m-separating.
S |S|
In Section 2, we make use of certain heredity properties of union-closed
families to prove that if in addition is union-closed it can be at most
S
(m+1)-separating. The main result of that section, Theorem 3, establishes
that for any n there is a unique (up to relabelling of vertices) n-separating
union-closed family of minimal weight.
3
In the third section, we use Theorem 3 together with Reimer’s Theorem
to obtain lower bounds on the weight of n-separating union-closed families
of size m for every realisable pair (m,n).
We construct families of examples showing these boundsare sharp up to
a multiplicative factor of 2+O 1 .
log2m
In the final section we consid(cid:16)er a ge(cid:17)neralisation of our original problem.
We define the l-fold weight of a family to be
S
A
w ( ) := | | .
l
S l
A∈S(cid:18) (cid:19)
X
The 0-fold weight of is just the size of , while the 1-fold weight is the
S S
weightw( )weintroducedearlier. Similarlytothel = 1case, wecanbound
S
w below for l 2 when is separating using a combination of Reimer’s
l
≥ S
Theorem and Theorem 3 together with some elementary arguments. Again
we provide constructions showing our bounds are the best possible up to a
multiplicative factor of 2+O(1/log m). As instant corollaries to our re-
2
sults in sections 3 and 4, we have for any l 1 sharp (up to a multiplicative
≥
constant) lower bounds on the expected number of sets in containing a
S
randomly selected l-tuple from Ω( ). These results are related to a gener-
S
alisation of the union-closed sets conjecture.
2 Separation
In this section we use our definition of separation to prove some results
about separating union-closed families. We begin with an item of notation.
Let be a family with domain Ω. Given X Ω, we will denote by [X]
S ⊆ S
the family induced by X on ,
S
[X] := A X A X,A .
S { \ | ⊇ ∈ S}
We shall consider [X] as a family with domain Ω( ) X. In a slight abuse
S S \
of notation we shall usually write [x] for [ x ]. Note that [x] = d (x).
S
S S { } |S |
Recall that separates a pair (i,j) of elements of Ω( ) if there exists
S S
A such that A contains exactly oneof i and j. is said to beseparating
∈ S S
if it separates every pair of distinct elements of Ω( ). We introduce an
S
equivalence relation = on its domain Ω( ) by setting x = y if does not
∼S S ∼S S
separate x from y. Quotienting Ω by = in the obvious way, we obtain a
∼S
reduced family
′ = / =
S S ∼S
on a newdomain Ω′ consisting of the= equivalence classes on Ω. It follows
∼S
from the definition of = that ′ is separating and uniquely determined by
∼S S
the knowledge of and Ω. We shall refer to ′ as the reduction of .
S S S
4
Union-closure is clearly preserved by our quotienting operation. Every
union-closed family S may thus be reduced to a unique separating union-
closed family in this way. Such separating union-closed families will be the
main object we study in this paper. Before proving anything about them,
let us give a few examples.
For n 2, we define the staircase of height n to be the union-closed
≥
family
T = n , n 1,n , n 2,n 1,n ,... 2,3,...n
n
{{ } { − } { − − } { }}
with domain Ω(T ) = 1,2,3......n . Note that T is n-separating, has
n n
{ }
size n 1 and that V(T ) = Ω(T ), since the element 1 is not contained in
n n
− 6
any set of T . For completeness, we define T to be the empty family with
n 1
domain Ω(T ) = 1 and size 0. Recall that T [X] is the subfamily of T
1 n n
{ }
induced by X. T has the property that T [ n ] = T .
n n n−1
{ } ∪{∅}
We shall prove that T is an n-separating union-closed family of least
n
weight.
For n 2, the plateau of width n is the n-separating union-closed family
≥
U = 1,2,...n 1 , 1,2,...n 2,n ,... 1,3,4...n , 2,3,...n ,[n] .
n
{{ − } { − } { } { } }
with domain Ω(U ) = [n] and size n+1. For completeness we let E be the
n 1
family , 1 withdomain 1 . ItiseasytoseethatU isthen-separating
n
{∅ { }} { }
union-closed family of sizen+1 withmaximal weight. Ithasweight roughly
twice that of T , and the additional property that for every pair i,j [n]
n
{ } ⊆
there is a set in U containing i and not j as well as a set containing j and
n
not i.
Finally, or n 1, the powerset of [n], P = [n] is, of course, a n-
n
≥ P
separating union-closed family with domain Ω(P ) = V(P ) = [n]. Note
n n
that P [ n ] = P , and that P is thelargest n-separating family in every
n n−1 n
{ }
sense of the word, having both the maximum size and the maximum weight
possible.
Let us now turn to the main purpose of this section.
We begin with a trivial lemma.
Lemma 1. Let be a separating family on Ω = [n] with elements labelled
S
in order of increasing degree. Then if 1 i< j n there exists A with
≤ ≤ ∈ S
j A, i / A.
∈ ∈
Proof. Since is separating, there is some A in containing one but not
S S
both of i, j. But we also know that d (i) d (j), so at least one such A
S S
≤
contains j and not i.
Repeated applications of Lemma 1 yield the following:
5
Lemma 2. Let be a separating union-closed family with Ω( ) = [n] and
S S
elements of Ω labelled in order of increasing degree. Then for every i
∈
[n 1], contains a set A = ([n] [i]) X , where X [i 1]. These n 1
i i i
− S \ ∪ ⊆ − −
sets are distinct.
Proof. Pick i [n 1]. By Lemma 1, for each j > i there exists B
j
∈ − ∈ S
containing j and not i. Let A = B . By union-closure, A . A is
i j>i j i ∈ S i
clearly of the form i+1,i+2,...n X , where X is a subset of [i 1].
{ S }∪ i i −
Moreover if i< j we have A = A since j A , j / A .
i j i j
6 ∈ ∈
The main result of this section follows easily.
Theorem 3. Let be a separating union-closed family on Ω( )= [n] with
S S
elements labelled in order of increasing degree. Then d (i) i 1 for all
S
≥ −
i [n]. In particular, n 1, and the weight of satisfies :
∈ |S|≥ − S
n
w( ) .
S ≥ 2
(cid:18) (cid:19)
Moreover, w( ) = n if and only if is one of T or T , where T is
S 2 S n n∪{∅} n
the staircase of height n introduced earlier.
(cid:0) (cid:1)
Proof. ByLemma2, containsn 1distinctsetsA ,A ,...A suchthat
1 2 n−1
S −
[n] [i] A . It follows in particular that n 1 and that d (i) i 1
i S
\ ⊆ |S|≥ − ≥ −
for all i [n]. Moreover
∈
w( ) A
i
S ≥ | |
i∈[n−1]
X
n
(n i) =
≥ − 2
i∈[n−1] (cid:18) (cid:19)
X
withequality ifandonlyifA = [n] [i]foreveryiandinaddition contains
i
\ S
no nonempty set other than the A . Thus w( ) = n if and only if is one
i S 2 S
of T or T , as claimed.
n n∪{∅} (cid:0) (cid:1)
3 Minimal weight
InthissectionweuseReimer’sTheoremandTheorem3togethertoobtaina
lower bound on the weight of an n-separating union-closed family of size m.
We then give constructions in the entire range of possible n, log m n
2 ≤ ≤
m+1, showing our bounds are asymptotically sharp except in the region
n = Θ mlog m (where they are differ by a multiplicative factor of at
2
(cid:0)p (cid:1)
6
most 2). As a corollary, we obtain a lower bound on the average degree in
a separating union-closed family.
Let be an n-separating union-closed family with = m. Recall that
S |S|
the weight of , w( ) is
S S
w( ) = A = d (x).
S
S | |
A∈S x∈Ω(S)
X X
We know from Reimer’s Theorem that
mlog m
w( ) 2 .
S ≥ 2
We have another bound for w( ) coming from our separation result, Theo-
S
rem 3:
n(n 1)
w( ) − .
S ≥ 2
If n 1 1+ 1+4mlog m = mlog m+O(1), the ‘bound in m’ from
≤ 2 2 2
Reimer’sTheoremisstronger;ifontheotherhandn 1 1+ 1+4mlog m ,
(cid:0) p (cid:1) p ≥ 2 2
the ‘bound in n’ from Theorem 3 is sharper.
(cid:0) p (cid:1)
For the bound in m, equality occurs if and only if is a powerset,
S
that is if and only n = log m. For the bound in n, equality occurs if and
2
only if is a staircase (with possibly the empty set added in). This can
S
only occur if n = m or n = m + 1. Remarkably the combined bound is
asymptotically sharp everywhere except in the region n = Θ mlog m ,
2
where it is only asymptotically sharp up to a constant. We shall show
(cid:0)p (cid:1)
this by constructing intermediate families between powersets and staircases.
Roughly speakingthese intermediary families will look like staircases sitting
on top of a powerset-like bases. This will allow Reimer’s Theorem and
Theorem 3 to give us reasonably tight bounds. Some technicalities arise to
make this work for all all possible (m,n).
We call apair of integers (n,m)satisfiable if thereexists an n-separating
union-closed family of size m – in particular n and m must satisfy n 1
− ≤
m 2n. Of course for m = 2n the powerset P is the only n-separating
n
≤
family of the right size. By Theorem 3 we know already how to construct
n-separating union-closed families of sizes m = n 1or m = n withminimal
−
weight. Alsoifm = n+1,itiseasytoseethatthefamilyT n 1
n
∪{∅}∪{{ − }}
has minimal weight, so for our purposes we may as well assume 2n > m >
n+1 in what follows.
Given asatisfiablepair(m,n)with2n > m > n+1,thereexists aunique
integer b such that 2b b m n < 2b+1 (b+1). Our aim is to take for
− ≤ − −
our powerset-like base a suitable family of m (n b 1) subsets of [b+1],
− − −
and to place on top of it a staircase of height n (b+1), thus obtaining a
−
separating union-closed family with the right size and domain.
For such a b we have 2b+1 m n+b+1 2b+1. Write out the binary
≤ − ≤
expansion of m n+b+1 as 2b1+2b2+...2bt with 0 b < b < ... < b ,
t t−1 1
− ≤
7
and note b b b + 1. We shall build the base of our intermediate
1
≤ ≤ B
family by adding up certain subcubes of [b+1].
P
First of all if b = b+ 1, we shall just let be the whole of [b +1].
1
B P
This is the “nontechnical case” of our construction. If on the other hand
b = b, we let Q denote theb -dimensional subcube X b+1 X [b] ,
1 1 1
{ ∪{ } | ⊆ }
and for every i : 2 i t we let Q be the b -dimensional subcube
i i
≤ ≤
X b X [b ] . We then set = Q .
{ ∪{ i−1}| ⊆ i } B i i
It is easy to see that the Q are disjoint. Indeed write b for b+1 and
i 0
S
suppose i < j; for every X Q , b is the largest element in X whereas
i i−1
∈
for every X′ Q , b < b is the largest element contained in X′, so
j j−1 i−1
∈
that X = X′.
6
Claim. is a (b+1)-separating union-closed family.
B
Proof. Q is (b+1)-separating since it contains the singleton b +1 and
1
{ }
the pairs i,b+1 for every i < b+1. Thus is (b+1)-separating also.
{ } B
Clearly each of the Q is closed under pairwise unions. Now consider
i
1 i< j (or alternatively b > b > b ) and take X Q , Y Q . Then
0 i j i j
≤ ∈ ∈
Y [b ] b
j j−1
⊆ ∪{ }
[b ],
i
⊆
from which it follows that X Y [b ] b , and hence that X Y Q .
i i−1 i
∪ ⊆ ∪{ } ∪ ∈
Thus = Q is closed under pairwise unions, as claimed.
B i i
We nowSturn to the staircase-like top of our family, , which we set to
T
be
= [b+2],[b+3],...[n] .
T { }
Our intermediate family will then be:
=
S B∪T
It is easy to see from our construction that is union-closed, n-separating
S
and has size
+ = (m n+b+1)+(n b 1) = m.
|B| |T| − − −
Wedonotclaim that isann-separatingunion-closedfamily ofsizemwith
S
minimal weight; however as we shall see w( ) is quite close to minimal.
S
Lemma 4.
log
w( ) < |B| 2|B| + .
B 2 |B|
8
Proof. Inthe“non-technicalcase”where = [b+1]ourassertionistrivial.
B P
We turn therefore to the “technical case” where = 2b1+2b2+2b3+...2bt
|B|
with b = b > b > ... > b 0:
1 2 t
≥
b
w( ) = 2bi i +1
B 2
i:Xbi6=0 (cid:18) (cid:19)
b b b+2
= 2bi + 2bi i−
2 2
i:Xbi6=0 i:Xbi6=0
b
|B| +2b1 +2b2/2
≤ 2
log
< |B| 2|B| + .
2 |B|
n(n+1)
Now m, and the weight of is clearly less than . Thus it
|B| ≤ T 2
follows that
mlog m n(n+1)
w( ) < 2 + +m.
S 2 2
On the other hand we already know from Reimer’s theorem and Theo-
rem 3 that
mlog m n(n 1)
w( ) max 2 , − ,
S ≥ 2 2
(cid:18) (cid:19)
whichisasymptotically thesameexceptwhenn2 mlog mwhenthelower
∼ 2
and upper bounds may diverge by a multiplicative factor of at most 2.
We have thus proved the following theorem.
Theorem 5. Let (n,m) be a satisfiable pair of integers. Suppose is an
S
n-separating union-closed family of size m with minimal weight. Then
mlog m n(n 1) mlog m n(n+1)
max 2 , − w( ) 2 + +m.
2 2 ≤ S ≤ 2 2
(cid:18) (cid:19)
In particular if (nm,m)m∈N is a sequence of satisfiable pairs and m a se-
S
quenceofn -separating union-closedfamiliesofsizemwithminimalweight,
m
we have the following:
If n /√mlogm 0 as m then
m
• → → ∞
mlog m
lim w( )/( 2 ) = 1.
m
m→∞ S 2
If n /√mlogm as m then
m
• → ∞ → ∞
n2
lim w( )/( ) = 1.
m
m→∞ S 2
9
Otherwise
•
n2 mlog m
1 lim w( )/max( , 2 ), and
m
≤ S 2 2
n2 mlog m
lim w( )/max( , 2 ) 2
m
S 2 2 ≤
As a corollary to Theorems 3, 5 and Reimer’s Theorem we have the
following result regarding average degree.
Corollary 6. Let be a separating union-closed family. Then,
S
1 log
d (x) |S| 2|S| +O(1).
S
Ω( ) ≥ 2
| S |x∈Ω(S) p
X
Moreover there exist arbitrarily large separating union-closed families with
1
d (x) log +O( /log ),
Ω( ) S ≤ |S| 2|S| |S| 2|S|
| S | x∈Ω(S)
X p p
so our bound is asymptotically sharp except for a multiplicative factor of at
most 2.
Proof. The average degree in a separating family is
S
1 w( )
d (x)= S .
S
Ω( ) Ω( )
| S | x∈Ω(S) | S |
X
If is an n-separating union-closed family of size m, we get two lower
S
bounds on w( ) from Reimer’s Theorem and Theorem 3. Dividing through
S
by Ω( ) = n and optimising yields
| S |
1 log 1
d (x) |S| 2|S| .
S
Ω( ) ≥ 2 − 4
| S | x∈Ω(S) p
X
The constructions from the proof of Theorem 5 then give us for each
satisfiable pair (n,m) examples of n-separating families of size m with close
to minimal average degree. In particular, take m = 2r and n= √2rr : the
⌈ ⌉
correspondingfamilyweconstructedhasweight2rr+O(2r). Ithastherefore
average degree √r2r +O( 2r/r) = mlog m+O( m/log m).
2 2
We believe our boundps are in fapct asymptoticallypsharp, and that the
constructions we gave in the proof of Theorem 5 are essentially the best
possible. We conjecture to that effect.
10