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On a conjecture of Füredi
European Journal of Combinatorics 49 (2015) 1–12
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European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
On a conjecture of Füredi István Tomon University of Cambridge, United Kingdom
article
abstract
info
Article history: Received 8 May 2014 Accepted 17 February 2015
Füredi that the Boolean lattice 2[n] can be partitioned conjectured n into ⌊n/2⌋ chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant α ≈ 0.8482 with the following property: for every ϵ > 0, if n [n] is sufficiently n large, the Boolean lattice 2 has a chain partition √ into ⌊n/2⌋ chains, each of them of size between (α − ϵ) n and
√
O( n/ϵ). We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset P of width w has a chain partition into w chains, each of size at most 2w|P | + 5, and it has a chain partition into w chains, where each chain has size at least 2w − 21 . © 2015 Elsevier Ltd. All rights reserved. |P |
1. Introduction The Boolean lattice 2[n] is the power set of [n] = {1, . . . , n} ordered by inclusion. A chain is a subset of pairwise comparable elements. By the well known theorem of Sperner [14] the minimum number n of chains 2[n] can be partitioned into is ⌊n/2⌋ . A symmetric chain in 2[n] is a chain c1 ⊂ · · · ⊂ ck with |ci | = |c1 | + i − 1 and |c1 | + |ck | = n. n The Boolean lattice 2[n] can be also partitioned into ⌊n/2⌋ symmetric chains (see [6]). This partition
contains l − l−1 chains of size n − 2l + 1 for l = 1, . . . , ⌊n/2⌋, so this partition contains ‘‘small’’ and ‘‘large’’ chains at the same time. The conjecture of Füredi [5] asks if there is a partition into the minimum number of chains such that the sizes of the chains are as close to each other as possible.
n
n
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.02.026 0195-6698/© 2015 Elsevier Ltd. All rights reserved.
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
n
Conjecture 1.1. For every positive integer n the Boolean lattice 2[n] can be partitioned into ⌊n/2⌋ chains n such that the length of each chain is l or l + 1, where l = ⌊2n / ⌊n/2⌋ ⌋.
√ is true, it means that there is a chain partition, where each chain has length √ If the conjecture ( π/ chain partition 2n+o(1)) n. Hsu, Logan, Shahriari and Towse [11,12] √ proved that there exists a √ into ⌊n/2⌋ chains such that the size of each chain is at least n/2+O(1), and at most O( n log n). Also, Hsu, Logan and Shahriari [10] proposed a more general conjecture; we believe that an even stronger conjecture holds as well. Before we state this conjecture, we recall some definitions concerning posets. These definitions can be found in [1], but we provide them here as well. Let (P , <) be a poset (partially ordered set). An antichain in P is a subset of pairwise incomparable elements. The width of P is the maximum size of an antichain in P. By the widely used theorem of Dilworth [4], the width of P is also the minimal number of chains needed to decompose P into chains. The poset P is graded if there exists a partition of its elements into subsets A0 , A1 , . . . , An such that A0 is the set of minimal elements, and whenever x ∈ Ai and x < y with no x < z < y, then y ∈ Ai+1 . If there exists such a partition, then it is unique and A0 , A1 , . . . , An are the levels of P. If x ∈ Ai , then the rank of x is i and it is denoted by rk(x). Clearly, 2[n] is graded with levels Ai = {x ∈ 2[n] : |x| = i}. The graded poset P is unimodal if there exists 0 ≤ m ≤ n such that |A0 | ≤ · · · ≤ |Am | and |Am | ≥ |Am+1 | ≥ · · · ≥ |An |. Also, P is rank-symmetric, if we have |Ai | = |An−i | for i = 0, . . . , n. A graded poset P is a normalized matching poset, if for every 1 ≤ i, j ≤ n and X ⊂ Ai we have
|X | |Γj (X )| ≤ , |Ai | |Aj | where Γj (X ) is the set of elements in Aj which are comparable with an element of X . Also, P is biregular, if for every 1 ≤ i, j ≤ n and x, y ∈ Ai we have |Γj ({x})| = |Γj ({y})|. It is easy to show that every biregular graded poset is a normalized matching poset. In particular, 2[n] is a unimodal normalized matching poset. Here is the promised strengthening of the conjecture of Hsu, Logan and Shahriari [10]. Conjecture 1.2. Let P be a unimodal normalized matching poset with width w . Then P can be partitioned into w chains, all of them of size l or l + 1, where l = ⌊|P |/w⌋. We note that the width of a normalized matching poset is just the size of the largest level (see Lemma 2.2). In [10], Conjecture 1.2 is stated for rank-symmetric, unimodal normalized matching posets. However, we believe this slightly more general conjecture is not harder than the original. In [10], this conjecture is proved for posets with 3 levels, and for finite linear lattices: the poset of the subspaces of the n-dimensional vector space over the finite field Fq (q is a prime power), ordered by inclusion. In this paper, we prove some results concerning the generalized conjecture and strengthen the results of [11,12]. Our paper is organized as follows. In the next section, we prove two theorems in the direction of Conjecture 1.2. The first theorem shows that if P is a unimodal normalized matching poset, then there is a chain partition into the minimal number of chains such that none of the chains are too large. Theorem 1.3. Let P be a unimodal normalized matching poset of width w . Then P can be partitioned into w chains, all of them of size at most 2w|P | + 5. In the second theorem we show that if P is a unimodal normalized matching poset, then there is a chain partition into the minimal number of chains such that every chain is large. Theorem 1.4. Let P be a unimodal normalized matching poset of width w . Then there exists a chain |P | partition into w chains such that each chain has size at least 2w − 12 . In Section 3, using our results from Section 2 we prove the following bounds for P = 2[n] .
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
3
Theorem 1.5. Let
α=
∞ √ √ log k −
2
k =2
√
log(k − 1)
k
≈ 0.8482.
For any ϵ > 0 there exists Nϵ such that if n > Nϵ , then 2[n] can be partitioned into ⌊n/2⌋ chains, all of √ √ them of size between (α − ϵ) n and O( n/ϵ).
n
In particular, one can get the following exact result by following the proof of Theorem 1.5. We shall avoid the calculations, we state the following corollary without proof.
n
Corollary 1.6. If n is sufficiently large, then 2[n] can be partitioned into ⌊n/2⌋ chains, all of them of size √ √ between 0.8 n and 13 n. In comparison, if Conjecture 1.1 is true, there is a chain partition with every chain having size √ ≈ 1.253 n. However, we are still far from proving that, Corollary 1.6 is an improvement of the results mentioned above. In Section 4, we prove a result for normalized matching posets, which are not necessarily unimodal. It turns out our methods used in Section 2 fail if do not suppose the unimodality of P. Instead, we shall use probabilistic tools to show that if the width of P is w , then there is a chain partition of P into
O(w log |P |) chains such that the size of each chain is at least Ω
|P | w log |P |
.
In the last section, we propose some open problems. 2. Unimodal normalized matching posets As mentioned before, in this section we prove Theorems 1.3 and 1.4. The proof of Theorem 1.3 is based on two simple lemmas, the first of which is about merging levels in normalized matchings. This lemma can be found in [10], but for completeness we state it here as well. First we need a definition. A bipartite graph G = (A, B; E ) is a normalized matching graph if for any X ⊂ A, Y ⊂ B we have
|X | |Γ (X )| ≤ |A| |B| and
|Y | |Γ (Y )| ≤ , |B| |A| where Γ (Z ) = ΓG (Z ) is the set of the neighbours of Z in G. Note that a graded poset is a normalized matching poset, if the bipartite graph induced by any two different levels in the comparability graph is a normalized matching graph. Lemma 2.1 ([10]). Let G1 = (A1 , B; E1 ), G2 = (A2 , B; E2 ) be normalized matching graphs with A1 and A2 disjoint. Let A = A1 ∪ A2 and E = E1 ∪ E2 . Then G = (A, B; E ) is also a normalized matching graph. Proof. First, let X ⊂ A, and X1 = X ∩ A1 , X2 = X ∩ A2 . Then
|ΓG1 (X1 ) ∪ ΓG2 (X2 )| |ΓG1 (X1 )| |ΓG2 (X2 )| |ΓG (X )| = ≥ max , |B| |B| |B| |B| |X1 | |X2 | | X1 | + | X2 | |X | , = . ≥ max ≥ |A1 | |A2 | |A1 | + |A2 | |A| Now let Y ⊂ B. Then
|ΓG (Y )| = |ΓG1 (Y )| + |ΓG2 (Y )| |A1 | |Y | |A2 | |Y | |A| |Y | ≥ + = . |B| |B| |B|
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Thus,
|Y | |ΓG (Y )| ≥ . |A| |B| The final lemma we need is just the well known result that any normalized matching poset has a chain partition, where the number of chains is only the size of the largest level. This can be found in [1] and we state it without proof. Lemma 2.2 ([1]). Let P be a normalized matching poset with levels A0 , . . . , An and m = max0≤i≤n |Ai |. Then the width of P is m, i.e. it can be partitioned into m chains. We are ready to prove Theorem 1.3. Proof of Theorem 1.3. Let A0 , . . . , An be the levels of P and set ai = |Ai |, i = 0, . . . , n. First, suppose that w = a0 ≥ a1 ≥ · · · ≥ an . Let d > 0 be the smallest positive integer satisfying ⌊ n/d⌋
ard ≤ w.
(1)
r =1
Claim 2.3. We have d < w + 1. |P |
Proof. By the monotonicity of the sequence {ai }ni=0 , for r = 1, . . . , ⌊n/d⌋ we have dard ≤ ard−d+1 +
· · · + ard . Hence, d
⌊ n/d⌋
d⌊n/d⌋
ard ≤
r =1
ai < |P |.
i =1
This means that (1) holds whenever d ≥ |P |/w . Thus, the smallest positive integer d satisfying (1) |P | satisfies d < w + 1. Define the graded poset Q as follows. Let its first d levels be A0 , . . . , Ad−1 and let its dth level be C =
⌊ n/d⌋
Ard .
r =1
Note that by the choice of d, |C | ≤ w . If x and y are elements in different levels of Q , then let x < y if x < y holds in P. Then Q is the subposet of P obtained by deleting all levels Ai with i > d and d ̸ |i, and merging all levels Aj with d|j. Thus, by repeated applications of Lemma 2.1, Q is also a normalized matching poset. Note that Q not need to be unimodal. By Lemma 2.2, Q has a partition into w chains. As Q has d + 1 levels, every chain has size at most d + 1. Let such a chain partition be {Di }w i=1 . For i = 1, . . . , ⌊n/d⌋, let Ri be the induced subposet of P
on the elements r =0 Aid+r (with Aj = ∅ if j > n). Then Ri is also a unimodal normalized matching poset with maximum sized level Aid . Thus, by Lemma 2.2 there exist chains {D′x }x∈Aid which partition Ri , |Dx | ≤ d and x is the minimum element of D′x for nall x ∈ Aid . Thus, the family {D′x }x∈C is a chain partition of j=d Aj such that each chain has size at most d and the minimum element of each chain is in C . For i = 1, . . . , w let Ti = Di if Di ∩ C = 0 and let Ti = Di ∪ D′x if Di ∩ C = {x}. Then Ti is also a chain as if Di ∩ C = {x}, then x is the maximum element of Di and the minimum element of D′x . Also |Ti | ≤ 2d as |D′x | ≤ d and |Di | ≤ d + 1. Thus, T1 , . . . , Tw is 2|P | a chain partition of P into w chains such that the length of each chain is at most 2d ≤ w + 2. This finishes the proof in the case |A0 | = w (see Fig. 1). Now suppose l > 0 is an index with w = |Al |. Let P1 be the subposet of P induced by the union of the levels A0 , . . . , Al and let P2 be induced by the union of the levels Al , . . . , An . By our previous results, 2|P | P1 can be partitioned into w chains {Cx }x∈Al with |Cx | ≤ w1 + 2, and P2 can be also partitioned into w
d−1
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
5
Fig. 1. P and Q .
chains {Cx′ }x∈Al satisfying |Cx′ | ≤ w2 + 2. But then, {Cx ∪ Cx′ }x∈Al is a chain partition of P into w chains satisfying 2|P |
|Cx ∪ Cx′ | = |Cx | + |Cx′ | − 1 ≤
2|P1 |
w
+
2|P2 |
w
+3=
2|P |
w
+ 5.
Next, we prove Theorem 1.4. Our proof uses the following simple and well known lemma about matchings, which can also be found in [2]. We state it without proof. Lemma 2.4 ([2]). Let G = (A, B; E ) be a bipartite graph. Suppose T is a matching in G. Then there exists a maximal matching (one with the maximum number of edges) that covers all the vertices covered by T . We shall put most of the work needed to prove Theorem 1.4 into the following technical lemma, which shall also be the main tool in the proof of Theorem 1.5. Lemma 2.5. Let P be a normalized matching poset with levels A0 , A1 , . . . , An and let ai = |Ai | for i = 0, . . . , n. Suppose that w = a0 ≥ a1 ≥ · · · ≥ an . Let f : {0, . . . , n} → N ∪ {∞} be defined by f (k) = min{i > k : ak+1 + · · · + ai ≥ w}, where ai = 0 if i > n. Also, let f1 = f and fi = f ◦ fi−1 denote the iterations of f ; set f0 ≡ 0. Finally, let d be the largest integer such that fd (0) < ∞. Then P can be partitioned into w chains, each of length at least d + 1. Proof. Let p = max{k : f (k) < ∞}. For i = 0, 1, . . . , n let A′i and A′′i be disjoint copies of Ai with bijections φi′ : A′i → Ai , φi′′ : A′′i → Ai and φi = φi′′−1 ◦ φi′ .
f (p)
p
Let A = i=0 A′i and B = j=1 A′′j . Define the bipartite graph G = (A ∪ B, E ) as follows: let x ∈ A′i , y ∈ A′′j , then xy ∈ E if i < j ≤ f (i) and φi′ (x) < φj′′ (y) holds in P. Suppose that there exists a complete matching from A to B in G. We know that there is a complete matching from A′′i to A′i−1 by the normalized matching property, so there is a complete matching from
p+1
B0 = j=1 A′′j to A. Thus, by Lemma 2.4, there is a complete matching from A to B which covers every element in B0 . Let M be such a matching. This matching corresponds to a family of w pairwise disjoint p+1 chains which cover i=0 Ai . Indeed, define the chain Cx for every x ∈ A0 by adding elements one by one. First, add the element x. Suppose that the last element we have added to Cx is y ∈ Ai . If i > p then stop. Otherwise, if φi′−1 (y) ∈ A is matched in M with some z ∈ A′′j , then add φj′′ (z ) to Cx . As the matching M covers B0
p+1
and A, the union x∈Ao Cx covers i=0 Ai . Also, |Cx | ≥ d + 1 for every x ∈ A0 . Indeed, if y < z are consecutive elements of Cx with y ∈ Ai and z ∈ Aj , then j ≤ f (i). Noting that f is strictly monotone increasing as a0 , . . . , an is monotone decreasing, the rank of the l-th smallest element of Cx is at most fl−1 (0), by induction. But the rank of the largest element of Cx is at least p + 1, so Cx has at least d + 1 elements. We show that these chains can be extended to form a partition of P. For i = p + 2, . . . , n, let Ni be a complete matching from Ai to Ai−1 . By the normalized matching property and Hall’s theorem [8] there exists a complete matching. Add the elements of P \ x∈A0 Cx one by one to one of the chains by the following procedure. Let y be an element with minimal rank, which is not contained in any of
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
the chains yet. Then rk(y) ≥ p + 2. Suppose y is matched to z in Nrk(y) and z is contained in the chain Cu for some u ∈ A0 . Then add y to Cu . By careful analysis of the procedure one can easily show that z is the largest element of Cu . Thus, Cu ∪ {y} is also a chain. Hence, when the procedure stops, we arrive at a chain partition of P into w chains, where every chain has size at least d + 1. Our task is reduced to showing that there is a complete matching from A to B. By Hall’s theorem it suffices to prove that for any X ⊂ A we have |ΓG (X )| ≥ |X |. For i = 0, . . . , p let Xi = X ∩ A′i and for any Z ⊂ A and j = 1, . . . , f (p) let Γj (Z ) = ΓG (Z ) ∩ A′′j . Then
p f (k) f (p) |ΓG (X )| = Γj (Xk ) = Γi (Xk ) k=0 j=k+1 i=1 k:k
f (p) i=1
max
k:k
|Γi (Xk )|.
By the normalized matching property the right hand side is at least f (p)
ai
i =1
max
|Xk |
k:k
ak
.
(2)
We want to show that (2) is at least |X |. We prove this inequality in a number of steps. For simplicity, write yi = |Xi |/ai . Introduce the indices L0 , L1 , . . . as follows. Let L0 = 0. If for i = 1, 2, . . . the index Li−1 is already defined and Li−1 < p, then let Li be any index j such that Li−1 < j ≤ p and yj is maximal among yLi−1 +1 , . . . , yp . Then L0 , L1 , . . . is a sequence of strictly monotone increasing integers not greater than p. Let s be the index such that Ls = p. Also, for r = 0, . . . , s define the sum Sr =
Lr
|Xi | + −a0 +
i=0
f (Lr )
f (p)
yLr +
ai
ai
i=f (Lr )+1
i=Lr +1
max
k:k
yk .
We shall show that S0 ≥ S1 ≥ · · · ≥ Ss . Suppose it holds. Note that S0 equals (2), so |Γ (X )| ≥ S0 . But then |Γ (X )| ≥ Ss , where Ss =
p
|Xi | + −a0 +
f (p)
ai
yp ≥
|Xi | = |X |.
i=0
i=p+1
i=0
p
Thus, |Γ (X )| ≥ |X |, and it completes the proof of this lemma. Our task is reduced to proving that the inequality Sr ≥ Sr +1 holds for r = 0, 1, . . . , s − 1. First of all, using that yLr ≥ yLr +1 and −a0 + Sr ≥
Lr
|Xi | + −a0 +
i =0
f (Lr )
i=Lr +1
f (Lr )
ai
yLr +1 +
i=Lr +1 f (p)
i=f (Lr )+1
ai ≥ 0, we have
ai
max
k:k
yk .
(3)
Let δ = f (Lr ) − Lr . If f (Lr ) + 1 ≤ i ≤ Lr +1 + δ then ai
max
k:k
yk ≥ ai yi−δ
= |Xi−δ | + (ai − ai−δ )yi−δ ≥ |Xi−δ | + (ai − ai−δ )yLr +1 .
(4)
The first inequality in (4) holds as f (k)−k is monotone increasing, so k = i−δ satisfies k < i ≤ f (k). The second inequality holds as ai − ai−δ ≤ 0 and yLr +1 ≥ yi−δ . Also, if Lr +1 + δ ≤ i ≤ f (Lr +1 ) then ai
max
k:k
yk = ai yLr +1 .
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
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Writing these bounds in (3) we deduce that Sr ≥
Lr
f (Lr )
|Xi | + −a0 +
i=0
ai
yLr +1
i=Lr +1 f (Lr +1 )
Lr +1 +δ
+
i=f (Lr )+1
=
Lr +1
|Xi−δ | + (ai − ai−δ )yLr +1 + f (Lr +1 )
|Xi | + −a0 +
f (p)
ai
yLr +1 +
ai
i=f (Lr +1 )+1
i=Lr +1 +1
f (p)
ai
i=f (Lr +1 )+1
i=Lr +1 +δ+1
i=0
ai yLr +1 +
max
k:k
max
k:k
yk
yk = Sr +1 .
Now we are ready to prove Theorem 1.4. Our task is reduced to showing that the d defined in the previous lemma is large in terms of |P |/w , and so applying Lemma 2.5 immediately gives the desired bound. Proof of Theorem 1.4. Let A0 , . . . , An be the levels of P. First, suppose that |A0 | = w . Define f and d as in Lemma 2.5. Note that f (k)
|Ai | ≤ 2w
i=k+1
holds for k = 0, 1, . . . , n, as by definition, f (k) is the smallest integer such that |Ai | ≤ w for i = 0, 1, . . . , n. Thus, for 0 ≤ l we have fl (0)
|Ai | = w +
l
fj (0)
i=k+1
|Ai | ≥ w , and
|Ai | ≤ (2l + 1)w.
j=1 i=fj−1 (0)+1
i=0
So, while |P | − (2l + 1)w ≥ w holds, we also have |P | − (2d + 1)w < w, which means we have d>
f (k)
|P | 2w
n
i=fl (0)+1
|Ai | ≥ w , meaning that d > l. Hence,
− 1.
Thus, by Lemma 2.5 it is immediate that P has a partition into w chains, all of them of size at least 2w . Now, suppose that |Am | = w with some m > 0. Let P1 be the normalized matching poset induced by the levels A0 , . . . , Am and let P2 be the poset induced by Am , . . . , An . Then P1 has a partition into w chains {Cx }x∈Am with |Cx | ≥ |P1 |/(2w) for all x ∈ Am and similarly, P2 has a partition into w chains {Cx′ }x∈Am with |Cx′ | ≥ |P2 |/(2w). Then {Cx ∪ Cx′ }x∈Am is a chain partition of P into w chains, each chain of size at least |P |
|P1 | |P2 | |P | 1 + −1= − . 2w 2w 2w 2 3. Boolean lattice [n] In this section, we prove Theorem 1.5. Observe that Theorem 1.4 immediately implies that 2 can n be partitioned into ⌊n/2⌋ chains of size at least
√ ( π /8 + o(1)) n. However, by estimating d given in Lemma 2.5 more carefully, we shall deduce an ever stronger bound. Proof of Theorem 1.5. The width of 2[n] is w = ⌊n/2⌋ . Let B0 , . . . , Bn be the levels of 2[n] . First, we need some approximations of the sizes of the levels close to the middle level.
n
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Using Stirling’s approximation one can easily get the estimation that for any t ∈ R we have
n n/2 + t
√
n
= (1 + o(1))2n e−2t
2
2
2
= (1 + o(1))e−2t w.
πn
(5)
For k = 1, 2, . . . let Tk > 0 be the smallest integer such that ⌈n/2⌉+T < w/k. Then, by (5) we k √ √ √ have Tk = (1/ 2 + o(1)) n log k. Let K be the smallest positive integer such that
α−
ϵ 2
<
K √ √ log k −
n
√
log(k − 1)
2
k
k=2
.
Note that for k > 2 we have
√
√
log k −
log(k − 1)
k Hence, the tail of the sum
=
∞
k=2
(log k) − (log(k − 1)) √ √
k( log(k − 1) + √
<
1 k2
.
√
log k− log(k−1) k
√ ∞ √ log k − log(k − 1) k
k=K +1
log k)
∞ 1
<
k2 k=K +1
can be easily bounded:
<
1 K
.
This implies K = O(1/ϵ). Let P be the subposet of 2[n] induced by the levels B⌈n/2⌉ , . . . , B⌈n/2⌉+TK and let Ai = B⌈n/2⌉+i for i = 0, 1, . . . , TK . Define f and d as in Lemma 2.5. For k = 2, 3, . . . , K , if Tk−1 < j ≤ Tk − k, then f (j) = j + k. Hence, d ≥
K Tk − Tk−1 − k k
k=2
≥
=
√ K √ √ 1 log k − log(k − 1) n √ + o(1) 2
k=2
k
√ 1 ϵ + o(1) n α− .
2
2
Thus,√ by Lemma 2.5 there exists a chain partition of P into w chains, each of size at least (1/2 + o(1)) n(α − ϵ/2) and at most
TK =
√ n log K . √ + o(1) 1
2
Let {Cx }x∈B⌈n/2⌉ be such a partition with x ∈ Cx for all x ∈ B⌈n/2⌉ . Let Q be the subposet of 2[n] induced by the levels B⌈n/2⌉+TK , . . . , Bn . Then Q is also a unimodal normalized matching poset with width
w = |B⌈n/2⌉+TK | = ′
1 K
+ o(1) w.
Thus, by Theorem 1.3, Q has a partition into w ′ chains, each of size at most 2|Q |
w′
+5=
2(K + o(1))|Q |
w
(K + o(1))2n πn < = (K + o(1)) . w 2
Let {Dy }y∈B⌈n/2⌉+T be such a partition with y ∈ Dy for all y ∈ B⌈n/2⌉+TK . K
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
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For all x ∈ B⌈n/2⌉ , let Cx′ = Cx if Cx ∩ B⌈n/2⌉+TK = ∅ and let Cx′ = Cx ∪ Dy if Cx ∩ B⌈n/2⌉+TK = {y}. Then {Cx }x∈B⌈n/2⌉ is a partition of the subposet R1 induced on {x ∈ 2[n] : |x| ≥ ⌈n/2⌉} with ′
ϵ√ πn + o(1) α − n < |Cx | < TK + (K + o(1)) 2 2 2 √ √ log K π n +K + o(1) . = n=O 2 2 ϵ 1
By symmetry, the subposet R2 induced on {x ∈ 2[n] : |x| ≤ ⌈n/2⌉} also has a chain partition into w chains satisfying the same bounds. Uniting the chain partitions of R1 and R2 in the obvious way (if two [n] chains have a common element take their √ to a chain partition of 2 such that each √ union), we arrive ϵ chain has size at least α − 2 + o(1) n and at most O n/ϵ . 4. Non-unimodal normalized matching posets In this section, we present a result on the chain partition of any normalized matching poset P. The proofs of Theorems 1.3 and 1.4 relied on unimodality, and we do not see any way to modify them for the non-unimodal case. For this case, we use a different approach, which however gives worse bounds. As usual, let w be the width of P, then we show that if |P |/w is not too small compared to |P |, then we can still show the existence of a chain partition of P into large chains. Theorem 4.1. Let P be a normalized matching poset (not necessarily unimodal) with |P | ≥ 3. Let the number of levels of P be n + 1, and let w be the width of P. Suppose that n + 1 < |P |2 /(1600w 2 (log |P |)3 ). Then P can be partitioned into at most 2w log |P | chains, each of size at least |P |/(40w log |P |). Our proof is probabilistic. We prepare the proof with the following definitions and lemmas. A maximal chain in a graded poset is a chain which intersects every level. The following lemma, which can be found in various sources [3,13], let us turn the set of maximal chains into a suitable probability space. Lemma 4.2. Let P be a normalized matching poset. Then P has a regular cover: there are maximal chains C1 , . . . , Cs (not necessarily distinct) such that for any x ∈ P the value r (x) = |{i ∈ [s] : x ∈ Ci }| depends only on the rank of x. Corollary 4.3. Let P be a normalized matching poset with levels A0 , . . . , An and suppose |P | > 3. Let C denote the set of maximal chains of P. Then there exists a function λ : C → R+ such that (C , 2C , λ) is a probability space, and if C ∈ C is randomly chosen according to λ then P (x ∈ C ) = 1/|Ark(x) | for all x ∈ P. Proof. Let C1 , . . . , Cs be a regular cover of P. For any C ∈ C let
λ(C ) = |{i ∈ [s] : Ci = C }|/s. Then λ suffices.
We also need the following special case of Hoeffding’s inequality [9].
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Lemma 4.4. Let X1 , . . . , Xn be independent random variables taking values in {0, 1} and let X = Then for any t > 0 we have
n
i=1
Xi .
2 P (EX − X > t ) < e−2t /n .
Proof of Theorem 4.1. Let C denote the set of maximal chains of P and define λ as in Corollary 4.3. Choose M = ⌈2w log |P |⌉ chains randomly and independently from C according to the probability measure λ. Let these chains be C1 , . . . , CM . We can suppose that M ≤ |P |, otherwise choosing |P | one element chains proves the theorem. For every x ∈ P let nx = |{i ∈ [M ] : x ∈ Ci }|. Let A be the event that nx > 0 for all x ∈ P. Then P (A) = P (∃x ∈ P : x ̸∈ Ci ; i = 1, . . . , M ) ≤
P (x ̸∈ Ci ; i = 1, . . . , M )
x∈P
=
1−
x∈P
M
1
|Ark(x) |
M 1 ≤ |P | 1 − ≤ |P |e−M /w < 1/3. w
For i = 0, . . . , n, let Li = 20w log |P |/|Ai | and let B be the event, that nx < Lrk(x) for all x ∈ P. Then P (B ) ≤
n
P (∃l1 < · · · < lLi : x ∈ Clj ; j = 1, . . . , Li )
i=0 x∈Ai
<
n
|Ai |
i=0
Using the inequality P (B ) <
n
M Li
|Ai |
i =0
1
Li M
|Ai |
Li
.
< (eM /Li )Li , we get eM
|Ai |Li
Li
.
Here, eM /(|Ai |Li ) < 1/2. Hence, we have P (B ) <
n
|Ai |2−20w(log |P |)/|Ai | ≤
n
|Ai |2−20 log |P | <
i=0
i =0
n |Ai | 1 ≤ . 2 |P | 3 i=0
Thus, P (A) > 2/3 and P (B ) > 2/3, which implies A ∩ B ̸= ∅. From now on fix a choice C1 , . . . , CM ∈ C satisfying 0 < nx < Lrk(x) for all x ∈ P. For any x ∈ P, choose randomly and uniformly an element of the set
{i ∈ [M ] : x ∈ Ci }, and let I (x, i) be the indicator random variable of the event that i is chosen for x. Namely, for x ∈ Ci we have I (x, i) = 1 if i is chosen from {i ∈ [M ] : x ∈ Ci }, and I (x, i) = 0 otherwise. Then P (I (x, i) = 1) = 1/nx for all x ∈ Ci . For i = 1, . . . , M let Di = {x ∈ Ci : I (x, Ci ) = 1}. Then D1 , . . . , DM is a chain partition of P. Let t = |P |/(40w log |P |). We show that with positive probability the inequality |Dk | > t holds for all k = 1, . . . , M. First of all,
E(|Dk |) =
x∈Ck
P (I (x, k) = 1) =
1 x∈Ck
Thus, P (|Dk | < t ) < P (E|Dk | − |Dk | > t ).
nx
>
n i =0
|Ai | = 2t . 20w log |P |
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
11
But |Dk | = x∈Ck I (x, k), where {I (x, k)}x∈Ci is a set of independent indicator random variables. Thus, by Hoeffding’s inequality
1 2 2 2 2 . P (E|Dk | − |Dk | > t ) < e−2t /(n+1) = e−|P | /(1600w (log |P |) (n+1)) < |P | Hence, P (∃k : |Dk | < t ) ≤
M
P (|Dk | < t ) <
k=1
M
|P |
≤ 1,
so P (∀k : |Dk | > t ) > 0. We showed that there exists a chain partition of P into ⌈2w log |P |⌉ chains such that the size of each chain is at least |P |/(40w log |P |). Let us compare the result of Theorem 4.1 with our previous results. In case P = 2[n] , log |P | is too large and the condition of the theorem is not satisfied. However, let P = [k]m with the usual pointwise ordering. Then P is the poset whose elements are
{(a1 , . . . , am ) : 1 ≤ a1 , . . . , am ≤ k} and (a1 , . . . , am ) ≤ (b1 , . . . , bm ) if ai ≤ bi holds for i = 1, . . . , m. Then P is a normalized matching poset (see [1]) with (k − 1)m + 1 levels. Regard m as a fix constant, and suppose that k goes to infinity. There exists a constant c (m), dependent only on m such that the width of P is (c (m) + o(1))km−1 . Hence, if k is sufficiently large given m, then the condition of Theorem 4.1 is satisfied and we have a chain partition of [k]m such that the size of each chain is at least
(1 + o(1))k . 40c (m)m log k In comparison, Theorem 1.4 shows the existence of a chain partition of [k]m , where each chain has size at least
(1 + o(1))k . 2c (m) So in this case, we lose an O(log k) factor. 5. Open problems We finish our paper with some open problems. Füredi’s conjecture and Conjecture 1.2 are still open. We also mention another generalization of Conjecture 1.1, proposed by Griggs [7]. Conjecture 5.1. Let n be a positive integer and w = ⌊n/2⌋ . For i = 1, . . . , w , let σi = n − 2l + 1, where n n 0 ≤ l ≤ ⌊n/2⌋ is the unique integer satisfying l−1 < i ≤ l+1 . Also, let µ1 ≥ · · · ≥ µw be positive integers such that µ1 + · · · + µw = 2n , and suppose
j i=1
µi ≤
j
n
σi
i =1
holds for j = 1, . . . , w . Then there exist chains C1 , . . . , Cw partitioning 2[n] such that |Ci | = µi for i = 1, . . . , w . We note that σ1 , . . . , σw are the sizes of the chains in the symmetric chain decomposition of 2[n] . Clearly, if µ1 , . . . , µw does not satisfy the conditions of Conjecture 5.1, then there is no chain partition of 2[n] , with chains of sizes µ1 , . . . , µw . Hence, if the conjecture is true, we have a characterization of all the partitions of 2n into w parts, which can be realized as the sizes of chains in a chain partition of 2[n] into w chains.
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
We also propose an open problem. By relaxing the definition of symmetric chain, we introduce the notion of semi-symmetric chain: let a chain {c0 , . . . , ck } ⊂ 2[n] with c0 ⊂ · · · ⊂ ck be semi-symmetric, if |ci | + |ck−i | = n holds for i = 0, . . . , k. We believe that a statement similar to Füredi’s conjecture holds, if we also demand that our chains are semi-symmetric.
n
Conjecture 5.2. Let n be a positive integer. Then 2[n] can be partitioned into ⌊n/2⌋ semi-symmetric chains such that the size of any two chains in the partition differs in at most two. So far, the author of this paper could only prove the following weaker result. h be a positive n Let integer and let n > N (h) be sufficiently large. Then 2[n] can be partitioned into ⌊n/2⌋ semi-symmetric chains, each of size at least h. This result is unpublished while the preparation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
I. Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987. A.S. Asratian, T.M.J. Denley, R. Häggkvist, Bipartite Graphs and their Applications, Cambridge Univ. Press, 1998. D.E. Daykin, L.H. Harper, D.B. West, Some remarks on normalized matching, J. Combin. Theory Ser. A 35 (1983) 301–308. R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. 51 (1) (1950) 161–166. Z. Füredi, Problem session, in: Kombinatorik Geordneter Mengen, Oberwolfach, BRD, 1985. C. Greene, D.J. Kleitman, Strong versions of Sperner’s theorem, J. Combin. Theory Ser. A 20 (1) (1976) 80–88. J.R. Griggs, Problems on chain partitions, Discrete Math. 72 (1988) 157–162. P. Hall, On representatives of subsets, J. Lond. Math. Soc. 10 (1) (1935) 26–30. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (301) (1963). T. Hsu, M.J. Logan, S. Shahriari, The generalized Füredi conjecture holds for finite linear lattices, Discrete Math. 306 (2006) 3140–3144. T. Hsu, M.J. Logan, S. Shahriari, C. Towse, Partitioning the Boolean lattice into chains of large minimum size, J. Combin. Theory Ser. A 97 (1) (2002) 62–84. T. Hsu, M.J. Logan, S. Shahriari, C. Towse, Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size, European J. Combin. 24 (2003) 219–228. D.J. Kleitman, On an extremal property of antichains in partial orders. The lym property and some of its implications and applications, in: NATO Advanced Study Institutes Series. Vol. 16, 1975, pp. 277–290. E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1) (1928) 544–548 (in German).
Contents lists available at ScienceDirect
European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
On a conjecture of Füredi István Tomon University of Cambridge, United Kingdom
article
abstract
info
Article history: Received 8 May 2014 Accepted 17 February 2015
Füredi that the Boolean lattice 2[n] can be partitioned conjectured n into ⌊n/2⌋ chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant α ≈ 0.8482 with the following property: for every ϵ > 0, if n [n] is sufficiently n large, the Boolean lattice 2 has a chain partition √ into ⌊n/2⌋ chains, each of them of size between (α − ϵ) n and
√
O( n/ϵ). We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset P of width w has a chain partition into w chains, each of size at most 2w|P | + 5, and it has a chain partition into w chains, where each chain has size at least 2w − 21 . © 2015 Elsevier Ltd. All rights reserved. |P |
1. Introduction The Boolean lattice 2[n] is the power set of [n] = {1, . . . , n} ordered by inclusion. A chain is a subset of pairwise comparable elements. By the well known theorem of Sperner [14] the minimum number n of chains 2[n] can be partitioned into is ⌊n/2⌋ . A symmetric chain in 2[n] is a chain c1 ⊂ · · · ⊂ ck with |ci | = |c1 | + i − 1 and |c1 | + |ck | = n. n The Boolean lattice 2[n] can be also partitioned into ⌊n/2⌋ symmetric chains (see [6]). This partition
contains l − l−1 chains of size n − 2l + 1 for l = 1, . . . , ⌊n/2⌋, so this partition contains ‘‘small’’ and ‘‘large’’ chains at the same time. The conjecture of Füredi [5] asks if there is a partition into the minimum number of chains such that the sizes of the chains are as close to each other as possible.
n
n
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.02.026 0195-6698/© 2015 Elsevier Ltd. All rights reserved.
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
n
Conjecture 1.1. For every positive integer n the Boolean lattice 2[n] can be partitioned into ⌊n/2⌋ chains n such that the length of each chain is l or l + 1, where l = ⌊2n / ⌊n/2⌋ ⌋.
√ is true, it means that there is a chain partition, where each chain has length √ If the conjecture ( π/ chain partition 2n+o(1)) n. Hsu, Logan, Shahriari and Towse [11,12] √ proved that there exists a √ into ⌊n/2⌋ chains such that the size of each chain is at least n/2+O(1), and at most O( n log n). Also, Hsu, Logan and Shahriari [10] proposed a more general conjecture; we believe that an even stronger conjecture holds as well. Before we state this conjecture, we recall some definitions concerning posets. These definitions can be found in [1], but we provide them here as well. Let (P , <) be a poset (partially ordered set). An antichain in P is a subset of pairwise incomparable elements. The width of P is the maximum size of an antichain in P. By the widely used theorem of Dilworth [4], the width of P is also the minimal number of chains needed to decompose P into chains. The poset P is graded if there exists a partition of its elements into subsets A0 , A1 , . . . , An such that A0 is the set of minimal elements, and whenever x ∈ Ai and x < y with no x < z < y, then y ∈ Ai+1 . If there exists such a partition, then it is unique and A0 , A1 , . . . , An are the levels of P. If x ∈ Ai , then the rank of x is i and it is denoted by rk(x). Clearly, 2[n] is graded with levels Ai = {x ∈ 2[n] : |x| = i}. The graded poset P is unimodal if there exists 0 ≤ m ≤ n such that |A0 | ≤ · · · ≤ |Am | and |Am | ≥ |Am+1 | ≥ · · · ≥ |An |. Also, P is rank-symmetric, if we have |Ai | = |An−i | for i = 0, . . . , n. A graded poset P is a normalized matching poset, if for every 1 ≤ i, j ≤ n and X ⊂ Ai we have
|X | |Γj (X )| ≤ , |Ai | |Aj | where Γj (X ) is the set of elements in Aj which are comparable with an element of X . Also, P is biregular, if for every 1 ≤ i, j ≤ n and x, y ∈ Ai we have |Γj ({x})| = |Γj ({y})|. It is easy to show that every biregular graded poset is a normalized matching poset. In particular, 2[n] is a unimodal normalized matching poset. Here is the promised strengthening of the conjecture of Hsu, Logan and Shahriari [10]. Conjecture 1.2. Let P be a unimodal normalized matching poset with width w . Then P can be partitioned into w chains, all of them of size l or l + 1, where l = ⌊|P |/w⌋. We note that the width of a normalized matching poset is just the size of the largest level (see Lemma 2.2). In [10], Conjecture 1.2 is stated for rank-symmetric, unimodal normalized matching posets. However, we believe this slightly more general conjecture is not harder than the original. In [10], this conjecture is proved for posets with 3 levels, and for finite linear lattices: the poset of the subspaces of the n-dimensional vector space over the finite field Fq (q is a prime power), ordered by inclusion. In this paper, we prove some results concerning the generalized conjecture and strengthen the results of [11,12]. Our paper is organized as follows. In the next section, we prove two theorems in the direction of Conjecture 1.2. The first theorem shows that if P is a unimodal normalized matching poset, then there is a chain partition into the minimal number of chains such that none of the chains are too large. Theorem 1.3. Let P be a unimodal normalized matching poset of width w . Then P can be partitioned into w chains, all of them of size at most 2w|P | + 5. In the second theorem we show that if P is a unimodal normalized matching poset, then there is a chain partition into the minimal number of chains such that every chain is large. Theorem 1.4. Let P be a unimodal normalized matching poset of width w . Then there exists a chain |P | partition into w chains such that each chain has size at least 2w − 12 . In Section 3, using our results from Section 2 we prove the following bounds for P = 2[n] .
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
3
Theorem 1.5. Let
α=
∞ √ √ log k −
2
k =2
√
log(k − 1)
k
≈ 0.8482.
For any ϵ > 0 there exists Nϵ such that if n > Nϵ , then 2[n] can be partitioned into ⌊n/2⌋ chains, all of √ √ them of size between (α − ϵ) n and O( n/ϵ).
n
In particular, one can get the following exact result by following the proof of Theorem 1.5. We shall avoid the calculations, we state the following corollary without proof.
n
Corollary 1.6. If n is sufficiently large, then 2[n] can be partitioned into ⌊n/2⌋ chains, all of them of size √ √ between 0.8 n and 13 n. In comparison, if Conjecture 1.1 is true, there is a chain partition with every chain having size √ ≈ 1.253 n. However, we are still far from proving that, Corollary 1.6 is an improvement of the results mentioned above. In Section 4, we prove a result for normalized matching posets, which are not necessarily unimodal. It turns out our methods used in Section 2 fail if do not suppose the unimodality of P. Instead, we shall use probabilistic tools to show that if the width of P is w , then there is a chain partition of P into
O(w log |P |) chains such that the size of each chain is at least Ω
|P | w log |P |
.
In the last section, we propose some open problems. 2. Unimodal normalized matching posets As mentioned before, in this section we prove Theorems 1.3 and 1.4. The proof of Theorem 1.3 is based on two simple lemmas, the first of which is about merging levels in normalized matchings. This lemma can be found in [10], but for completeness we state it here as well. First we need a definition. A bipartite graph G = (A, B; E ) is a normalized matching graph if for any X ⊂ A, Y ⊂ B we have
|X | |Γ (X )| ≤ |A| |B| and
|Y | |Γ (Y )| ≤ , |B| |A| where Γ (Z ) = ΓG (Z ) is the set of the neighbours of Z in G. Note that a graded poset is a normalized matching poset, if the bipartite graph induced by any two different levels in the comparability graph is a normalized matching graph. Lemma 2.1 ([10]). Let G1 = (A1 , B; E1 ), G2 = (A2 , B; E2 ) be normalized matching graphs with A1 and A2 disjoint. Let A = A1 ∪ A2 and E = E1 ∪ E2 . Then G = (A, B; E ) is also a normalized matching graph. Proof. First, let X ⊂ A, and X1 = X ∩ A1 , X2 = X ∩ A2 . Then
|ΓG1 (X1 ) ∪ ΓG2 (X2 )| |ΓG1 (X1 )| |ΓG2 (X2 )| |ΓG (X )| = ≥ max , |B| |B| |B| |B| |X1 | |X2 | | X1 | + | X2 | |X | , = . ≥ max ≥ |A1 | |A2 | |A1 | + |A2 | |A| Now let Y ⊂ B. Then
|ΓG (Y )| = |ΓG1 (Y )| + |ΓG2 (Y )| |A1 | |Y | |A2 | |Y | |A| |Y | ≥ + = . |B| |B| |B|
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Thus,
|Y | |ΓG (Y )| ≥ . |A| |B| The final lemma we need is just the well known result that any normalized matching poset has a chain partition, where the number of chains is only the size of the largest level. This can be found in [1] and we state it without proof. Lemma 2.2 ([1]). Let P be a normalized matching poset with levels A0 , . . . , An and m = max0≤i≤n |Ai |. Then the width of P is m, i.e. it can be partitioned into m chains. We are ready to prove Theorem 1.3. Proof of Theorem 1.3. Let A0 , . . . , An be the levels of P and set ai = |Ai |, i = 0, . . . , n. First, suppose that w = a0 ≥ a1 ≥ · · · ≥ an . Let d > 0 be the smallest positive integer satisfying ⌊ n/d⌋
ard ≤ w.
(1)
r =1
Claim 2.3. We have d < w + 1. |P |
Proof. By the monotonicity of the sequence {ai }ni=0 , for r = 1, . . . , ⌊n/d⌋ we have dard ≤ ard−d+1 +
· · · + ard . Hence, d
⌊ n/d⌋
d⌊n/d⌋
ard ≤
r =1
ai < |P |.
i =1
This means that (1) holds whenever d ≥ |P |/w . Thus, the smallest positive integer d satisfying (1) |P | satisfies d < w + 1. Define the graded poset Q as follows. Let its first d levels be A0 , . . . , Ad−1 and let its dth level be C =
⌊ n/d⌋
Ard .
r =1
Note that by the choice of d, |C | ≤ w . If x and y are elements in different levels of Q , then let x < y if x < y holds in P. Then Q is the subposet of P obtained by deleting all levels Ai with i > d and d ̸ |i, and merging all levels Aj with d|j. Thus, by repeated applications of Lemma 2.1, Q is also a normalized matching poset. Note that Q not need to be unimodal. By Lemma 2.2, Q has a partition into w chains. As Q has d + 1 levels, every chain has size at most d + 1. Let such a chain partition be {Di }w i=1 . For i = 1, . . . , ⌊n/d⌋, let Ri be the induced subposet of P
on the elements r =0 Aid+r (with Aj = ∅ if j > n). Then Ri is also a unimodal normalized matching poset with maximum sized level Aid . Thus, by Lemma 2.2 there exist chains {D′x }x∈Aid which partition Ri , |Dx | ≤ d and x is the minimum element of D′x for nall x ∈ Aid . Thus, the family {D′x }x∈C is a chain partition of j=d Aj such that each chain has size at most d and the minimum element of each chain is in C . For i = 1, . . . , w let Ti = Di if Di ∩ C = 0 and let Ti = Di ∪ D′x if Di ∩ C = {x}. Then Ti is also a chain as if Di ∩ C = {x}, then x is the maximum element of Di and the minimum element of D′x . Also |Ti | ≤ 2d as |D′x | ≤ d and |Di | ≤ d + 1. Thus, T1 , . . . , Tw is 2|P | a chain partition of P into w chains such that the length of each chain is at most 2d ≤ w + 2. This finishes the proof in the case |A0 | = w (see Fig. 1). Now suppose l > 0 is an index with w = |Al |. Let P1 be the subposet of P induced by the union of the levels A0 , . . . , Al and let P2 be induced by the union of the levels Al , . . . , An . By our previous results, 2|P | P1 can be partitioned into w chains {Cx }x∈Al with |Cx | ≤ w1 + 2, and P2 can be also partitioned into w
d−1
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
5
Fig. 1. P and Q .
chains {Cx′ }x∈Al satisfying |Cx′ | ≤ w2 + 2. But then, {Cx ∪ Cx′ }x∈Al is a chain partition of P into w chains satisfying 2|P |
|Cx ∪ Cx′ | = |Cx | + |Cx′ | − 1 ≤
2|P1 |
w
+
2|P2 |
w
+3=
2|P |
w
+ 5.
Next, we prove Theorem 1.4. Our proof uses the following simple and well known lemma about matchings, which can also be found in [2]. We state it without proof. Lemma 2.4 ([2]). Let G = (A, B; E ) be a bipartite graph. Suppose T is a matching in G. Then there exists a maximal matching (one with the maximum number of edges) that covers all the vertices covered by T . We shall put most of the work needed to prove Theorem 1.4 into the following technical lemma, which shall also be the main tool in the proof of Theorem 1.5. Lemma 2.5. Let P be a normalized matching poset with levels A0 , A1 , . . . , An and let ai = |Ai | for i = 0, . . . , n. Suppose that w = a0 ≥ a1 ≥ · · · ≥ an . Let f : {0, . . . , n} → N ∪ {∞} be defined by f (k) = min{i > k : ak+1 + · · · + ai ≥ w}, where ai = 0 if i > n. Also, let f1 = f and fi = f ◦ fi−1 denote the iterations of f ; set f0 ≡ 0. Finally, let d be the largest integer such that fd (0) < ∞. Then P can be partitioned into w chains, each of length at least d + 1. Proof. Let p = max{k : f (k) < ∞}. For i = 0, 1, . . . , n let A′i and A′′i be disjoint copies of Ai with bijections φi′ : A′i → Ai , φi′′ : A′′i → Ai and φi = φi′′−1 ◦ φi′ .
f (p)
p
Let A = i=0 A′i and B = j=1 A′′j . Define the bipartite graph G = (A ∪ B, E ) as follows: let x ∈ A′i , y ∈ A′′j , then xy ∈ E if i < j ≤ f (i) and φi′ (x) < φj′′ (y) holds in P. Suppose that there exists a complete matching from A to B in G. We know that there is a complete matching from A′′i to A′i−1 by the normalized matching property, so there is a complete matching from
p+1
B0 = j=1 A′′j to A. Thus, by Lemma 2.4, there is a complete matching from A to B which covers every element in B0 . Let M be such a matching. This matching corresponds to a family of w pairwise disjoint p+1 chains which cover i=0 Ai . Indeed, define the chain Cx for every x ∈ A0 by adding elements one by one. First, add the element x. Suppose that the last element we have added to Cx is y ∈ Ai . If i > p then stop. Otherwise, if φi′−1 (y) ∈ A is matched in M with some z ∈ A′′j , then add φj′′ (z ) to Cx . As the matching M covers B0
p+1
and A, the union x∈Ao Cx covers i=0 Ai . Also, |Cx | ≥ d + 1 for every x ∈ A0 . Indeed, if y < z are consecutive elements of Cx with y ∈ Ai and z ∈ Aj , then j ≤ f (i). Noting that f is strictly monotone increasing as a0 , . . . , an is monotone decreasing, the rank of the l-th smallest element of Cx is at most fl−1 (0), by induction. But the rank of the largest element of Cx is at least p + 1, so Cx has at least d + 1 elements. We show that these chains can be extended to form a partition of P. For i = p + 2, . . . , n, let Ni be a complete matching from Ai to Ai−1 . By the normalized matching property and Hall’s theorem [8] there exists a complete matching. Add the elements of P \ x∈A0 Cx one by one to one of the chains by the following procedure. Let y be an element with minimal rank, which is not contained in any of
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
the chains yet. Then rk(y) ≥ p + 2. Suppose y is matched to z in Nrk(y) and z is contained in the chain Cu for some u ∈ A0 . Then add y to Cu . By careful analysis of the procedure one can easily show that z is the largest element of Cu . Thus, Cu ∪ {y} is also a chain. Hence, when the procedure stops, we arrive at a chain partition of P into w chains, where every chain has size at least d + 1. Our task is reduced to showing that there is a complete matching from A to B. By Hall’s theorem it suffices to prove that for any X ⊂ A we have |ΓG (X )| ≥ |X |. For i = 0, . . . , p let Xi = X ∩ A′i and for any Z ⊂ A and j = 1, . . . , f (p) let Γj (Z ) = ΓG (Z ) ∩ A′′j . Then
p f (k) f (p) |ΓG (X )| = Γj (Xk ) = Γi (Xk ) k=0 j=k+1 i=1 k:k
f (p) i=1
max
k:k
|Γi (Xk )|.
By the normalized matching property the right hand side is at least f (p)
ai
i =1
max
|Xk |
k:k
ak
.
(2)
We want to show that (2) is at least |X |. We prove this inequality in a number of steps. For simplicity, write yi = |Xi |/ai . Introduce the indices L0 , L1 , . . . as follows. Let L0 = 0. If for i = 1, 2, . . . the index Li−1 is already defined and Li−1 < p, then let Li be any index j such that Li−1 < j ≤ p and yj is maximal among yLi−1 +1 , . . . , yp . Then L0 , L1 , . . . is a sequence of strictly monotone increasing integers not greater than p. Let s be the index such that Ls = p. Also, for r = 0, . . . , s define the sum Sr =
Lr
|Xi | + −a0 +
i=0
f (Lr )
f (p)
yLr +
ai
ai
i=f (Lr )+1
i=Lr +1
max
k:k
yk .
We shall show that S0 ≥ S1 ≥ · · · ≥ Ss . Suppose it holds. Note that S0 equals (2), so |Γ (X )| ≥ S0 . But then |Γ (X )| ≥ Ss , where Ss =
p
|Xi | + −a0 +
f (p)
ai
yp ≥
|Xi | = |X |.
i=0
i=p+1
i=0
p
Thus, |Γ (X )| ≥ |X |, and it completes the proof of this lemma. Our task is reduced to proving that the inequality Sr ≥ Sr +1 holds for r = 0, 1, . . . , s − 1. First of all, using that yLr ≥ yLr +1 and −a0 + Sr ≥
Lr
|Xi | + −a0 +
i =0
f (Lr )
i=Lr +1
f (Lr )
ai
yLr +1 +
i=Lr +1 f (p)
i=f (Lr )+1
ai ≥ 0, we have
ai
max
k:k
yk .
(3)
Let δ = f (Lr ) − Lr . If f (Lr ) + 1 ≤ i ≤ Lr +1 + δ then ai
max
k:k
yk ≥ ai yi−δ
= |Xi−δ | + (ai − ai−δ )yi−δ ≥ |Xi−δ | + (ai − ai−δ )yLr +1 .
(4)
The first inequality in (4) holds as f (k)−k is monotone increasing, so k = i−δ satisfies k < i ≤ f (k). The second inequality holds as ai − ai−δ ≤ 0 and yLr +1 ≥ yi−δ . Also, if Lr +1 + δ ≤ i ≤ f (Lr +1 ) then ai
max
k:k
yk = ai yLr +1 .
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
7
Writing these bounds in (3) we deduce that Sr ≥
Lr
f (Lr )
|Xi | + −a0 +
i=0
ai
yLr +1
i=Lr +1 f (Lr +1 )
Lr +1 +δ
+
i=f (Lr )+1
=
Lr +1
|Xi−δ | + (ai − ai−δ )yLr +1 + f (Lr +1 )
|Xi | + −a0 +
f (p)
ai
yLr +1 +
ai
i=f (Lr +1 )+1
i=Lr +1 +1
f (p)
ai
i=f (Lr +1 )+1
i=Lr +1 +δ+1
i=0
ai yLr +1 +
max
k:k
max
k:k
yk
yk = Sr +1 .
Now we are ready to prove Theorem 1.4. Our task is reduced to showing that the d defined in the previous lemma is large in terms of |P |/w , and so applying Lemma 2.5 immediately gives the desired bound. Proof of Theorem 1.4. Let A0 , . . . , An be the levels of P. First, suppose that |A0 | = w . Define f and d as in Lemma 2.5. Note that f (k)
|Ai | ≤ 2w
i=k+1
holds for k = 0, 1, . . . , n, as by definition, f (k) is the smallest integer such that |Ai | ≤ w for i = 0, 1, . . . , n. Thus, for 0 ≤ l we have fl (0)
|Ai | = w +
l
fj (0)
i=k+1
|Ai | ≥ w , and
|Ai | ≤ (2l + 1)w.
j=1 i=fj−1 (0)+1
i=0
So, while |P | − (2l + 1)w ≥ w holds, we also have |P | − (2d + 1)w < w, which means we have d>
f (k)
|P | 2w
n
i=fl (0)+1
|Ai | ≥ w , meaning that d > l. Hence,
− 1.
Thus, by Lemma 2.5 it is immediate that P has a partition into w chains, all of them of size at least 2w . Now, suppose that |Am | = w with some m > 0. Let P1 be the normalized matching poset induced by the levels A0 , . . . , Am and let P2 be the poset induced by Am , . . . , An . Then P1 has a partition into w chains {Cx }x∈Am with |Cx | ≥ |P1 |/(2w) for all x ∈ Am and similarly, P2 has a partition into w chains {Cx′ }x∈Am with |Cx′ | ≥ |P2 |/(2w). Then {Cx ∪ Cx′ }x∈Am is a chain partition of P into w chains, each chain of size at least |P |
|P1 | |P2 | |P | 1 + −1= − . 2w 2w 2w 2 3. Boolean lattice [n] In this section, we prove Theorem 1.5. Observe that Theorem 1.4 immediately implies that 2 can n be partitioned into ⌊n/2⌋ chains of size at least
√ ( π /8 + o(1)) n. However, by estimating d given in Lemma 2.5 more carefully, we shall deduce an ever stronger bound. Proof of Theorem 1.5. The width of 2[n] is w = ⌊n/2⌋ . Let B0 , . . . , Bn be the levels of 2[n] . First, we need some approximations of the sizes of the levels close to the middle level.
n
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Using Stirling’s approximation one can easily get the estimation that for any t ∈ R we have
n n/2 + t
√
n
= (1 + o(1))2n e−2t
2
2
2
= (1 + o(1))e−2t w.
πn
(5)
For k = 1, 2, . . . let Tk > 0 be the smallest integer such that ⌈n/2⌉+T < w/k. Then, by (5) we k √ √ √ have Tk = (1/ 2 + o(1)) n log k. Let K be the smallest positive integer such that
α−
ϵ 2
<
K √ √ log k −
n
√
log(k − 1)
2
k
k=2
.
Note that for k > 2 we have
√
√
log k −
log(k − 1)
k Hence, the tail of the sum
=
∞
k=2
(log k) − (log(k − 1)) √ √
k( log(k − 1) + √
<
1 k2
.
√
log k− log(k−1) k
√ ∞ √ log k − log(k − 1) k
k=K +1
log k)
∞ 1
<
k2 k=K +1
can be easily bounded:
<
1 K
.
This implies K = O(1/ϵ). Let P be the subposet of 2[n] induced by the levels B⌈n/2⌉ , . . . , B⌈n/2⌉+TK and let Ai = B⌈n/2⌉+i for i = 0, 1, . . . , TK . Define f and d as in Lemma 2.5. For k = 2, 3, . . . , K , if Tk−1 < j ≤ Tk − k, then f (j) = j + k. Hence, d ≥
K Tk − Tk−1 − k k
k=2
≥
=
√ K √ √ 1 log k − log(k − 1) n √ + o(1) 2
k=2
k
√ 1 ϵ + o(1) n α− .
2
2
Thus,√ by Lemma 2.5 there exists a chain partition of P into w chains, each of size at least (1/2 + o(1)) n(α − ϵ/2) and at most
TK =
√ n log K . √ + o(1) 1
2
Let {Cx }x∈B⌈n/2⌉ be such a partition with x ∈ Cx for all x ∈ B⌈n/2⌉ . Let Q be the subposet of 2[n] induced by the levels B⌈n/2⌉+TK , . . . , Bn . Then Q is also a unimodal normalized matching poset with width
w = |B⌈n/2⌉+TK | = ′
1 K
+ o(1) w.
Thus, by Theorem 1.3, Q has a partition into w ′ chains, each of size at most 2|Q |
w′
+5=
2(K + o(1))|Q |
w
(K + o(1))2n πn < = (K + o(1)) . w 2
Let {Dy }y∈B⌈n/2⌉+T be such a partition with y ∈ Dy for all y ∈ B⌈n/2⌉+TK . K
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
9
For all x ∈ B⌈n/2⌉ , let Cx′ = Cx if Cx ∩ B⌈n/2⌉+TK = ∅ and let Cx′ = Cx ∪ Dy if Cx ∩ B⌈n/2⌉+TK = {y}. Then {Cx }x∈B⌈n/2⌉ is a partition of the subposet R1 induced on {x ∈ 2[n] : |x| ≥ ⌈n/2⌉} with ′
ϵ√ πn + o(1) α − n < |Cx | < TK + (K + o(1)) 2 2 2 √ √ log K π n +K + o(1) . = n=O 2 2 ϵ 1
By symmetry, the subposet R2 induced on {x ∈ 2[n] : |x| ≤ ⌈n/2⌉} also has a chain partition into w chains satisfying the same bounds. Uniting the chain partitions of R1 and R2 in the obvious way (if two [n] chains have a common element take their √ to a chain partition of 2 such that each √ union), we arrive ϵ chain has size at least α − 2 + o(1) n and at most O n/ϵ . 4. Non-unimodal normalized matching posets In this section, we present a result on the chain partition of any normalized matching poset P. The proofs of Theorems 1.3 and 1.4 relied on unimodality, and we do not see any way to modify them for the non-unimodal case. For this case, we use a different approach, which however gives worse bounds. As usual, let w be the width of P, then we show that if |P |/w is not too small compared to |P |, then we can still show the existence of a chain partition of P into large chains. Theorem 4.1. Let P be a normalized matching poset (not necessarily unimodal) with |P | ≥ 3. Let the number of levels of P be n + 1, and let w be the width of P. Suppose that n + 1 < |P |2 /(1600w 2 (log |P |)3 ). Then P can be partitioned into at most 2w log |P | chains, each of size at least |P |/(40w log |P |). Our proof is probabilistic. We prepare the proof with the following definitions and lemmas. A maximal chain in a graded poset is a chain which intersects every level. The following lemma, which can be found in various sources [3,13], let us turn the set of maximal chains into a suitable probability space. Lemma 4.2. Let P be a normalized matching poset. Then P has a regular cover: there are maximal chains C1 , . . . , Cs (not necessarily distinct) such that for any x ∈ P the value r (x) = |{i ∈ [s] : x ∈ Ci }| depends only on the rank of x. Corollary 4.3. Let P be a normalized matching poset with levels A0 , . . . , An and suppose |P | > 3. Let C denote the set of maximal chains of P. Then there exists a function λ : C → R+ such that (C , 2C , λ) is a probability space, and if C ∈ C is randomly chosen according to λ then P (x ∈ C ) = 1/|Ark(x) | for all x ∈ P. Proof. Let C1 , . . . , Cs be a regular cover of P. For any C ∈ C let
λ(C ) = |{i ∈ [s] : Ci = C }|/s. Then λ suffices.
We also need the following special case of Hoeffding’s inequality [9].
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
Lemma 4.4. Let X1 , . . . , Xn be independent random variables taking values in {0, 1} and let X = Then for any t > 0 we have
n
i=1
Xi .
2 P (EX − X > t ) < e−2t /n .
Proof of Theorem 4.1. Let C denote the set of maximal chains of P and define λ as in Corollary 4.3. Choose M = ⌈2w log |P |⌉ chains randomly and independently from C according to the probability measure λ. Let these chains be C1 , . . . , CM . We can suppose that M ≤ |P |, otherwise choosing |P | one element chains proves the theorem. For every x ∈ P let nx = |{i ∈ [M ] : x ∈ Ci }|. Let A be the event that nx > 0 for all x ∈ P. Then P (A) = P (∃x ∈ P : x ̸∈ Ci ; i = 1, . . . , M ) ≤
P (x ̸∈ Ci ; i = 1, . . . , M )
x∈P
=
1−
x∈P
M
1
|Ark(x) |
M 1 ≤ |P | 1 − ≤ |P |e−M /w < 1/3. w
For i = 0, . . . , n, let Li = 20w log |P |/|Ai | and let B be the event, that nx < Lrk(x) for all x ∈ P. Then P (B ) ≤
n
P (∃l1 < · · · < lLi : x ∈ Clj ; j = 1, . . . , Li )
i=0 x∈Ai
<
n
|Ai |
i=0
Using the inequality P (B ) <
n
M Li
|Ai |
i =0
1
Li M
|Ai |
Li
.
< (eM /Li )Li , we get eM
|Ai |Li
Li
.
Here, eM /(|Ai |Li ) < 1/2. Hence, we have P (B ) <
n
|Ai |2−20w(log |P |)/|Ai | ≤
n
|Ai |2−20 log |P | <
i=0
i =0
n |Ai | 1 ≤ . 2 |P | 3 i=0
Thus, P (A) > 2/3 and P (B ) > 2/3, which implies A ∩ B ̸= ∅. From now on fix a choice C1 , . . . , CM ∈ C satisfying 0 < nx < Lrk(x) for all x ∈ P. For any x ∈ P, choose randomly and uniformly an element of the set
{i ∈ [M ] : x ∈ Ci }, and let I (x, i) be the indicator random variable of the event that i is chosen for x. Namely, for x ∈ Ci we have I (x, i) = 1 if i is chosen from {i ∈ [M ] : x ∈ Ci }, and I (x, i) = 0 otherwise. Then P (I (x, i) = 1) = 1/nx for all x ∈ Ci . For i = 1, . . . , M let Di = {x ∈ Ci : I (x, Ci ) = 1}. Then D1 , . . . , DM is a chain partition of P. Let t = |P |/(40w log |P |). We show that with positive probability the inequality |Dk | > t holds for all k = 1, . . . , M. First of all,
E(|Dk |) =
x∈Ck
P (I (x, k) = 1) =
1 x∈Ck
Thus, P (|Dk | < t ) < P (E|Dk | − |Dk | > t ).
nx
>
n i =0
|Ai | = 2t . 20w log |P |
I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
11
But |Dk | = x∈Ck I (x, k), where {I (x, k)}x∈Ci is a set of independent indicator random variables. Thus, by Hoeffding’s inequality
1 2 2 2 2 . P (E|Dk | − |Dk | > t ) < e−2t /(n+1) = e−|P | /(1600w (log |P |) (n+1)) < |P | Hence, P (∃k : |Dk | < t ) ≤
M
P (|Dk | < t ) <
k=1
M
|P |
≤ 1,
so P (∀k : |Dk | > t ) > 0. We showed that there exists a chain partition of P into ⌈2w log |P |⌉ chains such that the size of each chain is at least |P |/(40w log |P |). Let us compare the result of Theorem 4.1 with our previous results. In case P = 2[n] , log |P | is too large and the condition of the theorem is not satisfied. However, let P = [k]m with the usual pointwise ordering. Then P is the poset whose elements are
{(a1 , . . . , am ) : 1 ≤ a1 , . . . , am ≤ k} and (a1 , . . . , am ) ≤ (b1 , . . . , bm ) if ai ≤ bi holds for i = 1, . . . , m. Then P is a normalized matching poset (see [1]) with (k − 1)m + 1 levels. Regard m as a fix constant, and suppose that k goes to infinity. There exists a constant c (m), dependent only on m such that the width of P is (c (m) + o(1))km−1 . Hence, if k is sufficiently large given m, then the condition of Theorem 4.1 is satisfied and we have a chain partition of [k]m such that the size of each chain is at least
(1 + o(1))k . 40c (m)m log k In comparison, Theorem 1.4 shows the existence of a chain partition of [k]m , where each chain has size at least
(1 + o(1))k . 2c (m) So in this case, we lose an O(log k) factor. 5. Open problems We finish our paper with some open problems. Füredi’s conjecture and Conjecture 1.2 are still open. We also mention another generalization of Conjecture 1.1, proposed by Griggs [7]. Conjecture 5.1. Let n be a positive integer and w = ⌊n/2⌋ . For i = 1, . . . , w , let σi = n − 2l + 1, where n n 0 ≤ l ≤ ⌊n/2⌋ is the unique integer satisfying l−1 < i ≤ l+1 . Also, let µ1 ≥ · · · ≥ µw be positive integers such that µ1 + · · · + µw = 2n , and suppose
j i=1
µi ≤
j
n
σi
i =1
holds for j = 1, . . . , w . Then there exist chains C1 , . . . , Cw partitioning 2[n] such that |Ci | = µi for i = 1, . . . , w . We note that σ1 , . . . , σw are the sizes of the chains in the symmetric chain decomposition of 2[n] . Clearly, if µ1 , . . . , µw does not satisfy the conditions of Conjecture 5.1, then there is no chain partition of 2[n] , with chains of sizes µ1 , . . . , µw . Hence, if the conjecture is true, we have a characterization of all the partitions of 2n into w parts, which can be realized as the sizes of chains in a chain partition of 2[n] into w chains.
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I. Tomon / European Journal of Combinatorics 49 (2015) 1–12
We also propose an open problem. By relaxing the definition of symmetric chain, we introduce the notion of semi-symmetric chain: let a chain {c0 , . . . , ck } ⊂ 2[n] with c0 ⊂ · · · ⊂ ck be semi-symmetric, if |ci | + |ck−i | = n holds for i = 0, . . . , k. We believe that a statement similar to Füredi’s conjecture holds, if we also demand that our chains are semi-symmetric.
n
Conjecture 5.2. Let n be a positive integer. Then 2[n] can be partitioned into ⌊n/2⌋ semi-symmetric chains such that the size of any two chains in the partition differs in at most two. So far, the author of this paper could only prove the following weaker result. h be a positive n Let integer and let n > N (h) be sufficiently large. Then 2[n] can be partitioned into ⌊n/2⌋ semi-symmetric chains, each of size at least h. This result is unpublished while the preparation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
I. Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987. A.S. Asratian, T.M.J. Denley, R. Häggkvist, Bipartite Graphs and their Applications, Cambridge Univ. Press, 1998. D.E. Daykin, L.H. Harper, D.B. West, Some remarks on normalized matching, J. Combin. Theory Ser. A 35 (1983) 301–308. R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. 51 (1) (1950) 161–166. Z. Füredi, Problem session, in: Kombinatorik Geordneter Mengen, Oberwolfach, BRD, 1985. C. Greene, D.J. Kleitman, Strong versions of Sperner’s theorem, J. Combin. Theory Ser. A 20 (1) (1976) 80–88. J.R. Griggs, Problems on chain partitions, Discrete Math. 72 (1988) 157–162. P. Hall, On representatives of subsets, J. Lond. Math. Soc. 10 (1) (1935) 26–30. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (301) (1963). T. Hsu, M.J. Logan, S. Shahriari, The generalized Füredi conjecture holds for finite linear lattices, Discrete Math. 306 (2006) 3140–3144. T. Hsu, M.J. Logan, S. Shahriari, C. Towse, Partitioning the Boolean lattice into chains of large minimum size, J. Combin. Theory Ser. A 97 (1) (2002) 62–84. T. Hsu, M.J. Logan, S. Shahriari, C. Towse, Partitioning the Boolean lattice into a minimal number of chains of relatively uniform size, European J. Combin. 24 (2003) 219–228. D.J. Kleitman, On an extremal property of antichains in partial orders. The lym property and some of its implications and applications, in: NATO Advanced Study Institutes Series. Vol. 16, 1975, pp. 277–290. E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1) (1928) 544–548 (in German).