On the Fractional Dimension of Partially Ordered Sets
On the Fractional Dimension of
Partially Ordered Sets
STEFAN FELSNER
Bel l Communications Research South Street Morristown NJ U S A and Technische Uni
versit at Berlin Fachbereich Mathematik Strasse des Juni Berlin Germany
E mail address felsner math tu b erlin de
WILLIAM T TROTTER
Bel l Communications Research South Street L Morristown NJ U S A and Depart
ment of Mathematics Arizona State University Tempe AZ U S A
E mail address wtt b ellcore com
Date Septemb er
Abstract We use a variety of combinatorial techniques to prove several theorems concerning fractional
dimension of partially ordered sets In particular we settle a conjecture of Brightwell and Scheinerman
by showing that the fractional dimension of a p oset is never more than the maximum degree plus one
Furthermore when the maximum degree k is at least two we show that equality holds if and only if one
of the comp onents of the p oset is isomorphic to S the standard example of a k dimensional
k +1
p oset When w the fractional dimension of a p oset P of width w is less than w unless P contains
S If P is a p oset containing an antichain A and at most n other p oints where n we show that the
w
fractional dimension of P is less than n unless P contains S If P contains an antichain A such that all
n
antichains disjoint from A have size at most w then the fractional dimension of P is at most w and
this b ound is b est p ossible
Mathematics Sub ject Classi cation A C
Key words Partially p oset dimension fractional dimension degree
Intro duction
In this pap er we consider a partial ly ordered set P as a pair X P and refer to X as the
ground set and P as the partial order We nd it convenient to use the short form poset
for a partially ordered set and we use the term subposet to refer to a p oset induced by a
subset of the ground set
Let P X P b e a p oset and let F fM M g b e a multiset of linear extensions
1 t
of P Brightwell and Scheinerman call F a k fold realizer of P if for each incomparable
pair x y there are at least k linear extensions in F which reverse the pair x y i e
jfi i t x y in M gj k The fractional dimension of P denoted by fdim P is
i
then de ned in as the least real numb er q for which there exists a k fold realizer
F fM M g of P so that k t q it is easily veri ed that the least upp er b ound
1 t
of such real numb ers q is indeed attained Using this terminology the dimension of P
denoted by dim P is just the least t for which there exists a fold realizer of P It
follows immediately that fdim P dim P for every p oset P
In this pap er we will nd it convenient to use the following probabilistic interpretation
of this concept Let P X P b e a p oset and let F fM M g b e a multiset of
1 t
linear extensions of P We consider the linear extensions of F as outcomes in a uniform
sample space For an incomparable pair x y the probability that x is over y in F is
FRACTIONAL DIMENSION
given by
fi i t x y in M g Prob x y
i F
t
The fractional dimension of P is then the least rational numb er q so that there exists
a multiset F fM M g of linear extensions of P with Prob x y q for
1 t F
every incomparable pair x y
For each n the height two p oset containing n minimal elements fx x x g
1 2 n
and n maximal elements fy y y g with x y if and only if i j for all i j
1 2 n i j
n will b e denoted by S The p oset S is known as the standard example of a
n n
n dimensional p oset As noted in fdim S dim S n
n n
This pap er is organized as follows In Section we intro duce some terminology nec
essary for our theorems and review facts ab out lexicographic sums In Section we
formulate the results relating the fractional dimension of a p oset to its degree In Sec
tion we present a key technical lemma involving the algorithmic transformation of linear
orders into linear extensions and present the pro of for the main theorem of Section In
Section we present the forbidden subp oset characterization of the degree inequality In
Sections and we derive two other forbidden subp oset characterizations of inequalities
for fractional dimension In b oth cases equality is only p ossible when P contains a full
dimensional standard example as a subp oset
In Section we present an inequality relating the fractional dimension of a p oset
P X P to the width of the subp oset induced by X A where A is an antichain
The b ounds obtained in this case are slightly stronger than the corresp onding b ounds for
dimension We also prove that our b ounds are b est p ossible In Section we characterize
Hiraguchi s inequality for fractional dimension Finally in Section we prop ose some
new problems for fractional dimension
Notation and Terminology
Let P X P b e a p oset We denote the set of incomparable pairs of P by inc P Now
let F fM M M g b e a nonempty multiset of linear orders on X We call F a
1 2 t
multi realizer of P if P F i e each M is a linear extension of P and prob x y
i
F
for every x y inc P
When F is a multi realizer of P we de ne the value of F denoted value F as the
P
least rational numb er q so that prob x y q for every x y inc P The
F
fractional dimension of P is just the least rational q so that P has a multi realizer F
of value q
Recall that an incomparable pair x y in a p oset P X P is critical if any p oint
less than x is less than y and any p oint greater than y is greater than x We denote the set
of all critical pairs of P by crit P The following elementary prop osition rst noted by
Rabinovitch and Rival is stated formally to emphasize the fundamentally imp ortant
role played by critical pairs in dimension theory
PROPOSITION Let P X P be a poset and let F be a multiset of linear exten
sions of P Then F is a multi realizer of P if and only if prob x y for every
F
x y crit P
We say that a linear order L reverses the critical pair x y if x y in L Similarly we
say L reverses the set S crit P if L reverses each pair in S
The following remark is noted in
S FELSNER AND W T TROTTER
PROPOSITION Let P X P be poset and let q be a rational number Then
fdim P q if and only if there exists a multiset F of linear extensions of P so that
Prob x y q whenever x y crit P
F
Let P X P b e a p oset and let D fY Y Q x X g b e a family of p osets
x x x
indexed by the p oint set X of P The lexicographic sum of D over P is the p oset S whose
p oint set consists of all pairs of the form x y where x X and y Y The partial
x x
in S if and only if x y order on S is de ned by setting x y
2 x 1 x
2 1
x x in P or
1 2
in Q y x x and y
x x 1 2 x
1 2 1
Both dimension and fractional dimension are well b ehaved with resp ect to lexicographic
sums The statement for dimension is due to Hiraguchi and the statement for frac
tional dimension is given in
PROPOSITION Let S be the lexicographic sum of D fY x X g over P
x
X P Then
dim S maxfdim P max dim Y g
x X x
fdim S maxffdim P max fdim Y g
x X x
A p oset S is decomposable with resp ect to lexicographic sums if it is isomorphic to a
lexicographic sum of a family D fY x X g over a p oset P X P where jX j
x
and jY j for at least one x X Otherwise we say S is indecomposable
x
Given a p oint x in a p oset P X P let D x denote the set of all p oints which are
P
less than x in P and U x the set of p oints which are greater than x in P Whenever
P
the meaning is clear from the context we will drop the subscripts and just write D x
and U x Two p oints x and y are said to b e interchangeable in a p oset P X P if
D x D y and U x U y Thus x and y are interchangeable if and only if b oth
x y and y x are critical pairs Note that a p oset with an interchangeable pair is
decomp osable with resp ect to lexicographic sums unless it is a element antichain
Fractional Dimension and Degree
For a p oint x in a p oset P X P de ne the degree of x denoted deg x as the numb er
of p oints comparable but not equal to x in P Then let P denote the maximum degree
of P i e P maxfdeg x x X g
The following theorem is proved by Brightwell and Scheinerman in
THEOREM If P X P is a poset then fdim P P
In the next section we prove the following improved b ound which was conjectured in
THEOREM If P X P is a poset and is not an antichain then fdim P
P
As noted in fdim S n S for all n so Theorem is b est p ossible
n n
In Section we show that when the the maximum degree of a connected p oset P is
at least the inequality in Theorem is strict unless P is the standard example of
dimension P This result has much the same avor as Bro oks theorem which
asserts that the chromatic numb er of a connected graph is at most one more than the
maximum degree with equality holding if and only if the graph is a complete graph or an
o dd cycle It also provides another instance in which the role played by standard examples
FRACTIONAL DIMENSION
in dimension theory is analogous to the role played by complete graphs in the theory of
graph coloring This parallel is explored in greater detail in the monograph
We will in fact prove a result that is slightly stronger than Theorem Given a p oint
x in a xed p oset P X P let D x D x fxg Then let deg x jD x j and
D
de ne P maxfdeg x x X g The set U x and the quantities deg x P
D U
D U
are de ned dually
With this notation we can now state the result we actually prove
THEOREM If P X P is a poset then fdim P P
D
Note that Theorem has Theorem as an immediate corollary Since fractional
dimension is dual i e a p oset and its dual have the same fractional dimension the
following result will also b e an immediate corollary to Theorem
COROLLARY If P X P is a poset then fdim P minf P P g
D U
Linear Orders and Linear Extensions
In Furedi and Kahn showed that dimension theory problems could b e formulated in
terms of linear orders on the ground set instead of linear extensions Let P X P b e a
p oset and let L b e any linear order on the ground set X When S X and x X S we
write x S in L when x s in L for every s S We let C L f x y x y crit P
and x D y in Lg Furedi and Kahn noted that the following statement holds
PROPOSITION Let P X P be a poset and let L be any linear order on X Then
there exists a linear extension M of P with C L C M
Given a p oset P X P and a linear order L on X we let C L C L f x y x y
is a critical pair x D y in L and jD y j P g
D
LEMMA Let P X P be a poset with no interchangeable pairs and let L be any
linear order on X Then there exists a linear extension M of P with C L C M
Proof In order to simplify arguments to follow we present an algorithmic pro of We
describ e a deterministic algorithm LINEX which takes an arbitrary linear order L on X
as input and outputs a linear extension M LINEX L satisfying the conclusion of the
lemma
Set L L for j we obtain L from L by moving an element of X to an
0 j +1 j
alternate p osition according to rules which we describ e b elow The pro cedure halts when
we have a linear extension M L satisfying the conclusion of the lemma We divide the
j
pro cess into two phases The rst phase just transforms L into a linear extension N with
C L C N and C L C N so in this phase we will already b e proving a mo dest
extension of Prop osition
Phase Obtaining a Linear Extension This phase ends when L is a linear exten
j
sion of P If it is not among all pairs u v with u v in P and v u in L cho ose one
j
for which the numb er of p oints b etween them in L is minimum If there is more than one
j
such pair we cho ose one for which v is as low as p ossible in L
j
Claim If w is b etween u and v in L then w is incomparable with b oth u and v
j
Proof Supp ose to the contrary that w is b etween u and v in L but that w is compa
j
rable to u Now w u in L Also there are fewer p oints b etween w and u than b etween
j
u and v If u w in P then we contradict our choice of the pair u v as we would
S FELSNER AND W T TROTTER
prefer the pair u w On the other hand if w u in P then w v in P and we would
prefer v w to u v The argument when a p oint b etween u and v is comparable to v is
dual
Now form L from L by moving v to a p oint immediately over u From Claim
j +1 j
we note that L preserves exactly one more comparable pair of P than do es L This
j +1 j
insures that Phase will terminate
Claim C L C L
j j +1
Proof Supp ose to the contrary that there exists a critical pair x y C L C L
j j +1
and supp ose v was placed over u in the construction of L Then v D y v x in L
j +1 j
but x v in L However u v in P now implies that u y in P Now x y C L
j +1 j
requires x u in L This already gives x v in L a contradiction
j j +1
The argument for Claim also establishes the next claim which we state for emphasis
Claim C L C L
j j +1
We may now assume that Phase terminates with a linear extension N of P
Phase Doing More Set N N This phase pro duces a sequence fN j g of
0 j
linear extensions of P and terminates when M LINEX L N satis es C M C M
j
At each intermediate stage we will preserve the key prop erties that C N C N and
j j +1
C N C N
j j +1
Supp ose that there is some pair x y C N C N Cho ose one such pair and
j j
note that x y in N but z x in N for each z X with z y in P
j j
Claim No p oint b etween x and y in N is comparable to y
j
Proof Supp ose to the contrary that u is b etween x and y in N but that u is comparable
j
to y Since N is a linear extension of P we know that u y in P However this
j
contradicts the assumption that x y C L
Form the linear order N from N by moving y immediately b elow x In view of
j +1 j
Claim we know that N is a linear extension of P
j +1
Claim C N C N
j j +1
Proof Supp ose to the contrary that u v C N C N Then it follows that
j j +1
u y Since jD y j is maximum y is maximal element in P However since y v is a
critical pair we know D y D v and thus D y D v So v is also a maximal p oint
and U y U v We conclude that y and v are interchangeable
Exactly the same argument establishes the following claim
Claim C N C N
j j +1
Phase terminates b ecause C N is a prop er subset of C N as the second set
j j +1
contains x y but the rst do es not This completes the pro of of the lemma
We now present the pro of of Theorem
Proof Clearly we may assume that P is indecomp osable with resp ect to lexicographic
sums in particular P has no interchangeable pair
Let jX j n and set t n Then consider the set F fL L L g of all linear
1 2 t
orders on the ground set X Note that for a p oint x and a set S with x S the fraction of
linear orders in which x is higher than all p oints in S is exactly jS j It follows that
FRACTIONAL DIMENSION
a critical pair x y with jD y j k is in C L for n jD y j di erent p ermutations
L Also a critical pair x y with jD y j k is in C L for n jD y j di erent
p ermutations L
Apply the algorithm LINEX de ned in the previous section and let S fM M
1 2
M g where M LINEX L for each i t Lemma implies Prob x y
t i i S
jD y j when jD y j k and Prob x y jD y j when jD y j k
S
Thus Prob x y k for every critical pair x y in P
S
We remark that lo cal exchange arguments of the typ e used in this argument have also
b een applied in
Characterizing the Case of Equality
As noted by Brightwell and Scheinerman the standard example of an n dimensional p oset
satis es fdim P P so Theorem is b est p ossible In this section we will
show that the standard examples are the only connected p osets for which the inequality
in Theorem is tight Before presenting the details of the argument we need a little
more notation
Let P X P b e a p oset and let F b e a multi realizer of P with value F r We
de ne ess F f x y crit P prob x y r g The pairs in ess F are called
F
the essential pairs of F
We call a multi realizer F of a p oset P X P an optimal family for P if value F
fdim P Let F and F b e multi realizers of P We denote by F F the union of the
1 2 1 2
two multisets de ned so that if a linear order app ears n times in F and m times in F it
1 2
app ears n m times in F F The following elementary prop osition is stated formally
1 2
for emphasis
PROPOSITION If F and F are optimal multi realizers of a poset P then F F
1 2 1 2
is also an optimal multi realizer Furthermore
ess F F ess F ess F
1 2 1 2
We may then say that a critical pair x y crit P is an essential pair of P if x y
ess F for every optimal multi realizer F We let ess P denote the set of all essential
pairs of P Note that ess P is always nonempty when P is not a chain When L is a
linear order on X we let ess L C L ess P and ess L C L ess P The next
statement is an immediate consequence of the preceding de nitions
PROPOSITION Let P X P be a poset and let F be an optimal multi realizer
for P If M M F and ess M ess M then ess M ess M
1 2 1 2 1 2
We are now ready to present our characterization of Theorem
THEOREM If P X P is a poset with P then fdim P P
D D
unless P contains a standard example of a poset of dimension P
D
Proof Let P X P b e a p oset k P fdim P k and jX j n Without
D
loss of generality we may assume that P is indecomp osable with resp ect to lexicographic
sums in particular we may assume P has no interchangeable pair Again let S denote
the family of linear extensions of P constructed in the pro of of Theorem given in the
preceding section Then S is optimal We now derive some elementary statements ab out
the essential pairs reversed by a linear extension in S
S FELSNER AND W T TROTTER
Claim Let L b e any linear order and let M LINEX L Then ess L ess M
Proof Let x y crit P and let L b e a random linear order on X The probability
that x y b elongs to ess L is at least k Should this inequality not b e tight
then it follows that the pair x y do es not b elong to ess P If there is any linear order
L on X for which there exists a pair x y ess LINEX L ess L then we would
conclude that Prob x y k and thus x y ess P
S
Claim If x y ess P then jD y j P
D
Proof Supp ose to the contrary that r jD y j P k If r k then the
D
probability that x y C L for a random linear order on X is r which is greater
than k Hence x y ess P a contradiction
Now supp ose r k and let D y fy y y g Also let L b e any linear
1 2 k 1
order on X so that the lowest k elements of L are fxg D y with y y
1 2
y y x in L Let L b e obtained from L by switching the p ositions of x and y Then
k 1
x y ess L We conclude from Claim and Prop osition that there is a pair a b
in ess L ess L It is easy to see that this requires a b y z for some z which is
higher than x in L This is in turn requires deg z k and hence D z fxg D y
D
Let L b e obtained from L by moving y immediately ab ove x Now ess L $ ess L
r
which is a contradiction
Now x an essential pair y z of P and let D z fy y y g Then
k +1 k +1 k +1 1 2 k
let R b e any linear order on X in which the k lowest elements are fy y y g
1 2 k +1
with y y y in R Note that y z ess R
1 2 k +1 k +1 k +1
For each i k let R b e the linear order on X obtained by moving y im
i i
mediately over y in R Now cho ose an integer i f k g Then y z
k +1 k +1 k +1
ess R It follows that there is a pair in ess R ess R Claim implies that this pair
i i
has the form y z with jD z j k and hence D z fy y y y y g
i i i i 1 2 i 1 i+1 k +1
We conclude that the subp oset of P induced by fy y y g fz z z g is
1 2 k +1 1 2 k +1
a standard example of dimension k
We note that Theorem in is a sp ecial case of the preceding theorem We also
note that the characterization for Theorem follows immediately
COROLLARY If P X P is a poset with fdim P P then one
D
of the connected components of P is isomorphic to the standard example of dimension
P
D
Fractional Dimension and Width
The following inequality is due to Dilworth
THEOREM Let P X P be a poset Then dim P width P
The forbidden subp oset characterization problem for the inequality in Theorem remains
an unsolved and we susp ect very challenging problem although in some in nite
families of examples are constructed which must b e present in the solution Also it is not
known whether it is NP complete to determine whether the dimension of a p oset is less
than its width Using the techniques develop ed in the preceding sections we can answer
these questions completely for fractional dimension
Let P X P b e a p oset and let S and T b e disjoint subsets of X When M is a
linear extension of P we say S is over T in M and write S T in M when x y in M
for every x y inc P with x S and y T The following elementary result is due
to Hiraguchi
FRACTIONAL DIMENSION
PROPOSITION Let P X P be a poset and let C X be a chain Then there
exist linear extensions M M of P with C X C in M and X C C in M
1 2 1 2
Recall that Dilworth s theorem asserts that a p oset P of width w can b e partitioned
into w chains As a consequence the inequality in Theorem is an immediate corollary
to Prop osition
In what follows we will concentrate on chain decompositions also called chain covers
i e we will write the ground set as a union of not necessarily disjoint subsets each of
which is a chain
THEOREM Let P be a poset Then fdim P width P Furthermore fdim P
width P unless
width P or
width P w for some w and P contains the standard example of a t
dimensional poset as a subposet
Proof The rst statement is trivial Supp ose the second is false and cho ose a coun
terexample P X P with X as small as p ossible Let w width P fdim P
The minimality of jX j requires that fdim Q w for any prop er subp oset Q of P
Now supp ose that P has fewer than w maximal elements Then there exist a maximal
element y and a chain decomp osition X C C C of P so that y b elongs to two
1 2 w
or more of the chains in the decomp osition Let Q Y Q b e the subp oset obtained by
removing y from P and let fdim Q r w Let G b e an optimal realizer of Q and
supp ose that G consists of t linear extensions For each M G form a linear extension M
of X by adding y at the top of M Let G denote the resulting family of linear extensions
of P
Then for each i w let M b e a linear extension of P with X C C in
i i i
M Let F fM M M g and recall that value F w Then for each j
i 1 2 w
let j F denote the multi realizer of P consisting of j copies of each linear extension from
F
Let H G j F and observe that for j su ciently large value H w which is
j j
a contradiction
The same argument with an appropriate F shows that for each maximal element y of
P there is some x X so that x y ess P
Let fy y y g denote the set of maximal elements of P and cho ose an integer
1 2 w
i from f w g Then cho ose an element x X for which x y ess P Since
i i i
x y inc P we know that x C Cho ose an integer j j so that x C Then
i i i i i i j
x y We now show that x y in P for every k w with k i
i j i k
Supp ose to the contrary that there is some integer k f w g with k i for
which x y It follows easily that there is a linear extension M of P with X C C
i k k k
k
and x y in M However this implies that x y ess P The contradiction
i i i i
k
completes the pro of of our assertion that x y in P for every k w with
i k
k i
Now it follows that the p oints fx x x g form an antichain and the order induced
1 2 w
on fx x x g fy y y g is isomorphic to S
1 2 w 1 2 w w
The Complements of Antichains I
The following theorem was proved indep endently by Kimble and Trotter
S FELSNER AND W T TROTTER
THEOREM Let P X P be a poset let A X be an antichain and let n
jX Aj Then dim P maxf ng
In a forbidden subp oset characterization of this inequality is obtained The case
where jX Aj is trivial and the case jX Aj includes some pathology asso ciated
with the family of irreducible p osets However for jX Aj n there are exactly
n forbidden subp osets in the list
For fractional dimension the result is even more elegant
THEOREM Let P X P be a poset let A X be an antichain and let n
n 2
g unless jX Aj Then fdim P maxf ng Furthermore fdim P maxf n
n 1
n and P contains a element antichain or
n and P contains the standard example of an n dimensional poset as a sub
poset
Proof Again the rst statement is trivial We provide a sketch of the pro of of the
second omitting details for some routine parts of the argument When n it is proved
in that the dimension of P is at most n unless the n p oints of X A constitute
an antichain on one side of A that is completely ab ove or completely b elow A When
n we leave it to the reader to show that fdim P unless the p oints in X A
are an antichain and all three are on the same side of A Note that this statement is not
true for ordinary dimension
To complete the pro of it it su cient to show that the following p oset has fractional
dimension at most n n The antichain A is the set of minimal elements
of P The maximal elements of P are the n p oints in Y X A fy y y g
1 2 n
Furthermore for each nonempty subset S f ng with S f n g there
is an element a A with a y in P if and only if i S Note that by requiring
S S i
S f n g we are excluding the app earance of S as a subp oset of P
n
Here is the basic idea We describ e two families of linear extensions M fM M
1 2
M g and R fR R R g such that F M n R is a n fold realizer
n 1 2 n 1
of P
To de ne these extensions we rst describ e how they order the p oints of Y Once this
is done we have the gaps b etween these p oints into which the p oints of the antichain A
will b e inserted These insertions will b e done according to a sp eci ed set of rules Finally
we will provide a rule for ordering elements of A which are inserted into the same gap
Let a a note that a y in P if and only if i n Here are the rules for
n n i
fng
constructing M fM M M g
1 2 n
For each i n the p oint y is b elow all other elements of Y in M
i i
The largest element of Y in M is y
1 n
For each i n we insert a A fa g into the highest gap in M
n i
consistent with the partial order P
For each i n if a a A fa g b oth b elong to the same gap in M
n i
and jU a j jU a j then a a in M
i
The p oint a is the second largest element in M and the rst element of M for
n 1 i
each i n
Here are the rules for constructing R fR R R g
1 2 n 1
For each i n the p oint y is b elow all other elements of Y in R
i i
For each i n the p oint y is the second lowest element of Y in R
n i
FRACTIONAL DIMENSION
For each i n and each a A we insert a into the highest gap in M
i
consistent with the partial order P
For each i n if a a A b oth b elong to the same gap in R and
i
jU a j jU a j then a a in R
i
We leave it to the reader to verify that a critical pair of the form a a with n S
S n
is reverted in M for i n while every other critical pair x y is reversed in a
i
linear extension M M and in a linear extension R R This shows that F is a
2
n fold realizer Since F consists of n linear extensions we conclude that
fdim P n n as claimed
It is interesting to note that we have b een able to provide a discrete jump in the
fractional dimension in the characterization of equality in Theorem However there is
no such jump in the characterizations of equality for Theorem
REMARK For al l k and every there exists a poset P with P k
D
and k fdim P k
Let P b e the incidence p oset of a k regular k partite hyp ergraph where the sides if the
k partition all are of size n for some large n n
The Complements of Antichains II
There is another inequality b ounding the dimension of the complement of an antichain
This is another result of Trotter Let A X b e an antichain in P with X A
and let Q b e the subp oset of P induced by X A
THEOREM Let P X P be a poset with A and Q as above Then dim P
width Q
In it is shown that this inequality is b est p ossible The pro of makes use of the Product
Ramsey Theorem see Chapter of We now show that in the case of fractional
dimension the situation changes slightly For a p oset P X P and a subset Y X
we let P Y denote the restriction of P to Y so that Q Y P Y is the subp oset of P
induced by Y
THEOREM Let P X P be a poset let A be an antichain with X A nonempty
and let Q Y P Y Then let w width Q
If w or w then fdim P w
If w then fdim P w
If w then fdim P w
Furthermore for w and w the inequalities are asymptotical ly best possible
+
Proof Let C C C b e a chain partition of Q With C we denote the sub chain
1 2 w
i
+
of C ab ove A and with C the sub chain b elow A We may assume that C C C
i i
i i i
+
for all i Also let Y X A and de ne Y Y U A and Y Y D A
To prove the theorem we will construct a multirealizer F of P This multirealizer is
the union of three blo cks of linear extensions i e F B B B
1 2 3
Blo ck For each i w let L and M b e linear extensions with C X C and
i i i i
X C C Let L fL L g and M fM M g De ne B as pL pM
i i 1 w 1 w 1
the value of p p w to b e sp eci ed later
S FELSNER AND W T TROTTER
Blo ck In B we have for each i w a subblo ck C of linear extensions i e
2 i
+ +
+
B C C Each linear extension in C has Y C C and C Y C
2 1 w i
i i i i
Moreover for every subset B of A with jB j jAj we may assume jAj to b e even there
is a linear extension L in C in which elements of B are as low as p ossible while elements
B i
jAj
denote the numb er of linear extensions of A B are as high as p ossible Let q
jAj 2
in C
i
Blo ck In B we assemble r copies of a linear extension which simultaneously reverses
3
all critical pairs x y with x y A Assuming that there are no interchangeable pairs
this can easily b e done
To count how often a critical pair x y of P is reversed in F we have to distinguish
three typ es
Typ e x C and y C The pair is reversed whenever C X C and when
i j i i
X C C Since i j this gives p reversals in B
j j 1
Typ e x C and y A The pair is reversed whenever C X C giving p reversals
i i i
in B However x C and the pair is also reversed in a linear extension L C when
1 B i
i
y B Together this gives p q reversals By the dual argument the same numb er
counts the reversals when x A and y C
i
Typ e x A and y A The pair is reversed r times in B Also each C gives more
3 i
than q reversals together this makes more than r w q
Note that jF j p q w r We now give the prop ortions of p and r relative to q
w p r q
w p q r q
w p q r q
w p q r
A simple calculation shows that the multirealizers built with these prop ortions have
the right values Note that in the rst three cases the values p and r where chosen as to
balance the prop ortion of reversals of Typ e with the prop ortion of reversals of Typ es
and In the calculation we only counted w q reversals of Typ e in B As already
3
noted the actual numb er is slightly larger This proves that the given b ounds can not b e
attained
We now sketch the argument showing that for every there is a p oset P containing
an antichain A such that width P A w and the fractional dimension of P is at least
w
For given w and n let P X P b e the p oset de ned by
A is an antichain in P
C C is a chain partition of X A
1 w
+
jC j jC j n for all i
i i
+ + +
C is incomparable to C C is incomparable to C and C C for all i j
i j i j i j
For every down set S D A and every up set T U A there is an element
a A with D a S and U a T
S T S T S T
Now let F b e an optimal multirealizer of P in what follows we consider F as a probability
space We cho ose a and round all probabilities to multiples of Consider an
+
antichain B consisting of one p oint from C for each i For each in the symmetric
i
group on w ob jects let p b e the rounded probability that in a linear extension in F the
FRACTIONAL DIMENSION
1
w !
elements of B app ear in the order sp eci ed by There are fewer than p ossible
functions p and we color the antichains by these functions Provided n is large the
Pro duct Ramsey Theorem guarantees the existence of a two element sub chain E in each
i
+
C such that all antichains consisting of one p oint from each E receive the same color
i
i
Note that this implies that in all but a negligible p ortion of linear extensions in F the
sub chains E b ehave as p oints relative to one another i e if x E y E and x y
i i j
then E E
i j
The same argument allows us to cho ose two element sub chains F in C for each i
i
i
such that the F b ehave as p oints relative to each other almost all linear extensions in F
i
Now let a A b e the element with a max E ajj min E ajj max F and a min F
i i i i
for all i w Observe that when the E and F b ehave as p oints in a linear
i i
extension L then only one of the w events a min E a max F for i w
i i
can hold in L Hence the probability of the least probable of these events is at most
w where the probability that the E or the F do not b ehave as p oints
i i
only dep ends on the accuracy of the rounding i e on Now can b e chosen so that
w w This completes the pro of
Note that the examples given in the pro of of Theorem have already b een used in
to show that the inequality of Theorem is tight
Hiraguchi s Inequality and Removal Theorems
We now give two further characterizations for inequalities involving fractional dimension
Both results will b e easy corollaries of previous results and b oth show that some problems
are much easier when stated in terms of fractional dimension The rst example is known
as Hiraguchi s inequality In this case when jX j the characterizing p osets are
the same for b oth ordinary dimension and fractional dimension For ordinary dimension
the pro of is only mo derately complicated when jX j is even see However for ordinary
dimension there is no transparent pro of when jX j is o dd The most frequently cited source
for the pro of is Kimble but to the b est of our knowledge the arguments therein are
incomplete
By way of contrast the pro of of the characterization for fractional dimension is very
easy in b oth cases for an analogous situation see the characterization given for interval
dimension in
THEOREM If P X P and jX j then fdim P bjX j c Furthermore if
jX j then fdim P bjX j c only if jX j is even and P is a standard example or jX j
is odd and P contains a standard example on jX j points
Proof If P contains an antichain A with jAj jX j then Theorem implies
fdim P jX Aj jX j Now assume that there is no antichain of size more then
jX j then width P bjX j c In this case Theorem implies fdim P bjX j c
The characterization is an immediate consequence of the characterization of Theorem
THEOREM If P X P is a poset and x X then fdim X x P X x
fdim P Furthermore fdim X x P X x fdim P for each x X only if
P is a standard example
Note that the condition dim X x P X x dim P for each x X gives a
de nition for irreducible p osets A characterization of irreducible p osets has only b een
obtained for dimensions two and three An indication for the abundancy of irreducible
S FELSNER AND W T TROTTER
p osets is the following estimate For each t if n t then the numb er of t
irreducible p osets on n p oints is at least as large as the numb er of p osets on n p oints
having dimension at most t
Proof of Theorem In it is shown that a p oset P X P on n elements contains
n 2
fdim P Hence when fdim P an element x such that fdim X x P X x
n
jX j there is an element x X such that fdim X x P X x fdim P The
result now follows from Theorem
Op en Problems for Fractional Dimension
In most instances we have b een able to show that our inequalities for fractional dimension
are b est p ossible In two cases there is still some uncertainty
PROBLEM Are the fol lowing two inequalities of Theorem best possible
Let P X P be a poset let A be an antichain with X A nonempty and let
Q Y P Y Then let w width Q
If w then fdim P w
If w then fdim P w
For the inequality ofTheorem see the remarks at the end of Section we make
the following conjecture
CONJECTURE
For each there is an integer w so that for every w w there exists a poset
P so that w fdim P w width P
For every positive integer w there is an so that fdim P w for
w w
every poset P with fdim P w width P
Our results and techniques suggest that many dimension theoretic problems have analo
gous versions for fractional dimension which may b e somewhat more tractable Here are
two such problems The rst problem is just a restatement of a problem rst p osed by
Peter Fishburn see
PROBLEM For each t let f t be the minimum number of incomparable pairs
2
in a poset P with fdim P t Is it true that f t t
We note that the chevron has dimension and has only incomparable pairs
However an easy exercise shows that a p oset with fractional dimension at least must
have at least incomparable pairs The case t was resolved by Qin He showed
that a p oset must have at least incomparable pairs in order to have dimension
PROBLEM Given rational numbers p and q what is the minimum value of the
fractional dimension of P Q where fdim P p and fdim Q q
We were unable to make any progress on determining whether a p oset on three or more
p oints always contains a pair whose removal decreases the fractional dimension by at
most As is the situation with ordinary dimension this app ears to b e a very di cult
problem Perhaps the following restricted version is accessible
PROBLEM Does there exist an absolute constant so that any poset with or
more points always contains a pair whose removal decreases the fractional dimension by
at most
FRACTIONAL DIMENSION
Acknowledgement
The authors would like to express their appreciation to Peter Winkler for numerous stim
ulating conversations on the topics discussed in this pap er He also gave a careful reading
to a preliminary version and made many helpful suggestions for improving the exp osition
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