Sperners Theorem and its generalizations Matthias Beck Xueqin Wang - - PDF document

Sperner’s Theorem and its generalizations Matthias Beck Xueqin Wang Thomas Zaslavsky SUNY Binghamton www.binghamton.edu/matthias

“Everything should be made as simple as possible, but not simpler.” Albert Einstein “The worst thing you can do to a problem is solve it completely.” Daniel Kleitman

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Matroid theory Ehrhart theory Department Lattice!

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Define a weak partial composition into p parts as an ordered p -tuple

• A1, . . . , Ap
• such that A1, . . . , Ap are pairwise disjoint

sets. Theorem Suppose

• Aj1, . . . , Ajp
• for j =

1, . . . , m are different weak set compositions into p parts with the condition that, for all 1 ≤ k ≤ p and all I ⊆ [m] with |I| = r+1 , there exist distinct i, j ∈ I such that either Aik = Ajk or Aik ∩

• l=k

Ajl = ∅ = Ajk ∩

• l=k

Ail . Then

m

• j=1

1 |Aj1|+···+|Ajp|

|Aj1|,...,|Ajp|

≤ rp and m is bounded by the sum of the rp largest p -multinomial coefficients for inte- gers less than or equal to max

1≤j≤m

• |Aj1| + · · · + |Ajp|
• .

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Let S be an n-element set. Sperner’s Theorem (1928) Suppose A1, . . . , Am ⊆ S such that Ak ⊆ Aj for k = j. Then m ≤

• n

⌊n/2⌋

• .

LYM Inequality (Lubell, Yamamoto, Meshalkin, 1960±6) Suppose A1, . . . , Am ⊆ S such that Ak ⊆ Aj for k = j. Then

m

• k=1

1 n

|Ak|

≤ 1 . Both bounds can be attained for any n.

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Theorem (Erd˝

• s, 1945)

Suppose {A1, . . . , Am} ⊆ P(S) contains no chains with r + 1 elements. Then m is bounded by the sum of the r largest bino- mial coefficients n

k

• , 0 ≤ k ≤ n.

Theorem (Rota–Harper, 1970) Suppose {A1, . . . , Am} ⊆ P(S) contains no chains with r + 1 elements. Then

m

• k=1

1 n

|Ak|

≤ r . Both bounds can be attained for any n and r.

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Theorem (Griggs–Stahl–Trotter, 1984) Suppose

• Aj0, . . . , Ajq
• are m different chains

in P(S) such that Aji ⊆ Akl for all i and l and all j = k. Then m ≤

• n − q

⌊(n − q)/2⌋

• .

Theorem (Bollob´ as, 1965) Suppose

• Aj, Bj
• are m pairs of sets such

that Aj∩Bj = ∅ for all j and Aj∩Bk = ∅ for all j = k. Then

m

• j=1

1 |Aj|+|Bj| |Aj| ≤ 1 . Both bounds can be attained for any n and q.

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Define a weak composition of S into p parts as an ordered p -tuple A =

• A1, . . . , Ap
• f

sets Ak such that A1, . . . , Ap are pairwise disjoint and A1 ∪ · · · ∪ Ap = S . Theorem (Meshalkin, 1963) Suppose M = {A1, . . . , Am} is a class of weak compositions of S into p parts such that for all 1 ≤ k ≤ p the set {Aj

k}m j=1

forms an antichain. Then m = |M| is boun- ded by the largest p -multinomial coefficient for n. Theorem (Hochberg–Hirsch, 1970)

• A∈M

1

• n

|A1|,...,|Ap|

≤ 1 . Both bounds can be attained for any n and p.

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• Proof. p = 2: LYM.

For general p, let M(F) = {(A2, . . . , Ap) : (F, A2, . . . , Ap) ∈ M}. Then

• A∈M

1

• n

|A1|,...,|Ap|

=

• A∈M

1 n

|A1|

• n−|A1|

|A2|,...,|Ap|

• =
• F∈M1

1 n

|F|

• A′∈M(F)

1

• n−|F|

|A2|,...,|Ap|

• where A′ = (A2, . . . , Ap),

≤

• F∈M1

1 n

|F|

· 1 by the induction hypothesis, ≤ 1 by LYM.

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Sperner E GST M

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Theorem Suppose M = {A1, . . . , Am} is a class of weak compositions of S into p parts such that for all 1 ≤ k < p the set {Aj

k : 1 ≤ j ≤ m}

contains no chain of length r. Then (a)

• A∈M

1

• n

|A1|,...,|Ap|

≤ rp−1 (b) m = |M| is bounded by the sum of the rp−1 largest p -multinomial coefficients for n.

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Projective geometry P n−1(q) Rank of a flat r(a) = dim a + 1 q-Gaussian coefficients n k

• q =

n!q k!q(n − k)!q , where n!q = (qn − 1)(qn−1 − 1) · · · (q − 1) Theorem (Rota–Harper, 1970) Suppose {a1, . . . , am} ⊆ P n−1(q) contains no chains with r + 1 elements. (a)

m

• j=1

1

• n

r(aj)

• q

≤ r (b) m is at most the sum of the r largest Gaussian coefficients n

j

• q for 0 ≤ j ≤ n

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A Meshalkin sequence of length p in P n−1(q) is a sequence a = (a1, . . . , ap) of flats whose join is ˆ 1 and whose ranks sum to n . If a is a Meshalkin sequence, we write r(a) = (r(a1), . . . , r(ap)) for the sequence of ranks. For α = (α1, . . . , αp) we write s2(α) =

• i<j

αiαj and define the (q-)Gaussian multinomial co- efficient as n α

• q =

n!q α1!q · · · αp!q .

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Theorem Suppose M = {a1, . . . , am} is a family of Meshalkin sequences of length p in P n−1(q) such that for all 1 ≤ k < p the set

• aj

k : 1 ≤ j ≤ m

• contains no chain of length r . Then

(a)

• a∈M

1 n

r(a)

• q qs2(r(a)) ≤ rp−1

(b) m = |M| is at most equal to the sum of the rp−1 largest amongst the quantities n α

• q qs2(α)

for α = (α1, . . . , αp) with all αk ≥ 0 and α1 + · · · + αp = n

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Corollary Suppose M = {a1, . . . , am} is a family of Meshalkin sequences of length p in P n−1(q) such that for all 1 ≤ k < p the set

• aj

k : 1 ≤ j ≤ m

• is an antichain. Then

(a)

• a∈M

1 n

r(a)

• q qs2(r(a)) ≤ 1

(b) m = |M| ≤ max

α

n α

• q qs2(α)

(c) The bounds in (a) and (b) can be achieved for any n and p .

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A “stranger” LYM inequality is Corollary Suppose M = {a1, . . . , am} is a family of Meshalkin sequences of length p in P n−1(q) such that for all 1 ≤ k < p the set

• aj

k : 1 ≤ j ≤ m

• contains no chain of length r . Then
• a∈M

1 n

r(a)

• q

is bounded by the sum of the rp−1 largest expressions qs2(α) for α = (α1, . . . , αp) with all αk ≥ 0 and α1 + · · · + αp = n.

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