Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: - - PowerPoint PPT Presentation

Honors Combinatorics

CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 6, Thursday, May 14, 2020

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs

Theorem (Erd˝

• s)

If an r-uniform hypergraph has ≤ 2r−10 edges then it is 2-colorable.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs

Theorem (Erd˝

• s)

If an r-uniform hypergraph has ≤ 2r−10 edges then it is 2-colorable.

• Proof. Let the edges be E1, . . . , Em. Color the vertices

red/blue at random. Let Bi be the event that Ei becomes monochromatic (bad event). P(Bi) = 2 2r ∴ P( coloring illegal ) = P(m

i=1 Bi) ≤ m i=1 P(Bi) = 2m 2r ≤ 1 512

Not only does a good coloring exist, but 99.8% of the colorings works.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs QUESTION (Erd˝

• s)

What if we don’t limit the number of edges, only the degree of the vertices?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs QUESTION (Erd˝

• s)

What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs QUESTION (Erd˝

• s)

What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability? Close to 2r ?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs QUESTION (Erd˝

• s)

What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability? Close to 2r ?

YES

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs Random coloring? Suppose degree ≤ 1 (edges are disjoint) — Probability of success? ∴ P( coloring legal ) =

• 1 − 2

2r m < e−2m/2r → 0 as m → ∞ exponentially small. Random coloring will not work.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs Random coloring? Suppose degree ≤ 1 (edges are disjoint) — Probability of success? ∴ P( coloring legal ) =

• 1 − 2

2r m < e−2m/2r → 0 as m → ∞ exponentially small. Random coloring will not work. Lovász (1976): “Not so fast. Exponentially small but positive chance is still success.”

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = (V, F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event Bv

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = (V, F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event Bv Assumptions:

• 1. each Bv independent of the set

{Bw | w not a neighbor of v}

• 2. (∀v ∈ V)(deg(v) ≤ d)
• 3. (∀v ∈ V)(P(Bv) ≤ 1/(4d))

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = (V, F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event Bv Assumptions:

• 1. each Bv independent of the set

{Bw | w not a neighbor of v}

• 2. (∀v ∈ V)(deg(v) ≤ d)
• 3. (∀v ∈ V)(P(Bv) ≤ 1/(4d))

Conclusion: P       

• v∈V

Bv        > 0 We have positive chance to avoid all the bad events.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Independence from a set of events Let ǫ1, . . . , ǫk ∈ {1, −1}. Events B1, . . . , Bk define 2k atoms B ǫ1

1 ∩ · · · ∩ B ǫk k

where B1

i = Bi and B−1 i

= Bi.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Independence from a set of events Let ǫ1, . . . , ǫk ∈ {1, −1}. Events B1, . . . , Bk define 2k atoms B ǫ1

1 ∩ · · · ∩ B ǫk k

where B1

i = Bi and B−1 i

= Bi. DEF: Event A and the set {B1, . . . , Bk} of events are independent if A is independent of each atom of the Bi.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Independence from a set of events Let ǫ1, . . . , ǫk ∈ {1, −1}. Events B1, . . . , Bk define 2k atoms B ǫ1

1 ∩ · · · ∩ B ǫk k

where B1

i = Bi and B−1 i

= Bi. DEF: Event A and the set {B1, . . . , Bk} of events are independent if A is independent of each atom of the Bi. DO: A and {B1, . . . , Bk} are independent ⇐⇒ for every Boolean function f : {0, 1}k → {0, 1}, A is independent of f(B1, . . . , Bk).

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Conditional probabilities DEF: If P(B) > 0 then P(A | B) := P(A ∩ B) P(B)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Conditional probabilities DEF: If P(B) > 0 then P(A | B) := P(A ∩ B) P(B) Lemma P(A | C ∩ D) = P(A ∩ C | D) P(C | D)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Conditional probabilities DEF: If P(B) > 0 then P(A | B) := P(A ∩ B) P(B) Lemma P(A | C ∩ D) = P(A ∩ C | D) P(C | D)

• Proof. P(A | C ∩ D) = P(A ∩ C ∩ D)

P(C ∩ D) = P(A ∩ C | D)P(D) P(C | D)P(D)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = ([n], F ) graph; events Bi (i ∈ [n])

• 1. each Bi independent of the set

{Bj | j not a neighbor of i}

• 2. (∀i ∈ [n])(deg(i) ≤ d)
• 3. (∀i ∈ [n])
• P(Bi) ≤ 1

4d

• CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

Lovász Local Lemma G = ([n], F ) graph; events Bi (i ∈ [n])

• 1. each Bi independent of the set

{Bj | j not a neighbor of i}

• 2. (∀i ∈ [n])(deg(i) ≤ d)
• 3. (∀i ∈ [n])
• P(Bi) ≤ 1

4d

• Then P

n

i=1 Bi

• > 0

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = ([n], F ) graph; events Bi (i ∈ [n])

• 1. each Bi independent of the set

{Bj | j not a neighbor of i}

• 2. (∀i ∈ [n])(deg(i) ≤ d)
• 3. (∀i ∈ [n])
• P(Bi) ≤ 1

4d

• Then P

n

i=1 Bi

• > 0

Proof by induction on n. Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma G = ([n], F ) graph; events Bi (i ∈ [n])

• 1. each Bi independent of the set

{Bj | j not a neighbor of i}

• 2. (∀i ∈ [n])(deg(i) ≤ d)
• 3. (∀i ∈ [n])
• P(Bi) ≤ 1

4d

• Then P

n

i=1 Bi

• > 0

Proof by induction on n. Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

Condition has positive probability by inductive hypothesis

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma Theorem (∀i ∈ [n])

• P(Bi) ≤ 1

4d

• =⇒

P n

i=1 Bi

• > 0

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma Theorem (∀i ∈ [n])

• P(Bi) ≤ 1

4d

• =⇒

P n

i=1 Bi

• > 0

Proof by induction on n. Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma Theorem (∀i ∈ [n])

• P(Bi) ≤ 1

4d

• =⇒

P n

i=1 Bi

• > 0

Proof by induction on n. Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d Lemma =⇒ Theorem

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma Theorem (∀i ∈ [n])

• P(Bi) ≤ 1

4d

• =⇒

P n

i=1 Bi

• > 0

Proof by induction on n. Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d Lemma =⇒ Theorem

Proof: F := B2 ∩ · · · ∩ Bn. P(B1∩· · ·∩Bn) = P(B1∩F) = P(B1 | F)P(F) ≥

• 1 − 1

2d

• P(F) > 0

QED

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma

Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma

Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

• Proof. Induction on n. Neighbors of vertex 1:

{2, . . . , k} C := B2 ∩ · · · ∩ Bk D := Bk+1 ∩ · · · ∩ Bn P      B1

• n
• i=2

Bi       = P(B1 | C ∩ D) = P(B1 ∩ C | D) P(C | D)

?

≤ 1 2d

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma

Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

• Proof. Induction on n. Neighbors of vertex 1:

{2, . . . , k} C := B2 ∩ · · · ∩ Bk D := Bk+1 ∩ · · · ∩ Bn P      B1

• n
• i=2

Bi       = P(B1 | C ∩ D) = P(B1 ∩ C | D) P(C | D)

?

≤ 1 2d

Numerator: P(B1 ∩ C | D) ≤ P(B1 | D) = P(B1) ≤ 1 4d

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma

Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

• Proof. Induction on n. Neighbors of vertex 1:

{2, . . . , k} C := B2 ∩ · · · ∩ Bk D := Bk+1 ∩ · · · ∩ Bn P      B1

• n
• i=2

Bi       = P(B1 | C ∩ D) = P(B1 ∩ C | D) P(C | D)

?

≤ 1 2d

Numerator: P(B1 ∩ C | D) ≤ P(B1 | D) = P(B1) ≤ 1 4d Denominator: P(C | D) = 1 − P(C | D) = 1 − P(k

i=2 Bi | D)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Lovász Local Lemma

Lemma P      B1

• n
• i=2

Bi       ≤ 1 2d

• Proof. Induction on n. Neighbors of vertex 1:

{2, . . . , k} C := B2 ∩ · · · ∩ Bk D := Bk+1 ∩ · · · ∩ Bn P      B1

• n
• i=2

Bi       = P(B1 | C ∩ D) = P(B1 ∩ C | D) P(C | D)

?

≤ 1 2d

Numerator: P(B1 ∩ C | D) ≤ P(B1 | D) = P(B1) ≤ 1 4d Denominator: P(C | D) = 1 − P(C | D) = 1 − P(k

i=2 Bi | D)

≥ 1 − k

i=2 P(Bi | D) ≥ 1 − k i=2 1 2d ← by Inductive Hyp ≥ 1 2

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

2-colorable hypergraphs

Prove that every r-uniform hypergraph with maximum degree ≤ f(r) is 2-colorable, for some function f(r) that is close to 2r.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Kronecker product of matrices

A ∈ Fk×ℓ B ∈ Fr×s A ⊗ B ∈ Fkr×ℓs A ⊗ B =                a11B a12B . . . a1ℓB a21B a22B . . . a2ℓB . . . . . . . . . ak1B ak2B . . . akℓB                Ex. (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) If square matrices: trace(A ⊗ B) = trace(A) · trace(B) eigenvalues of A: λ1, . . . , λk eigenvalues of B: µ1, . . . , µr DO: eigenvalues of A ⊗ B: λiµj

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DEF A ∈ Mn(R) Hadamard matrix (1893): hij = ±1, columns orthogonal A ∈ Mn(±1) =⇒ (columns orthog ⇐⇒ rows orthog)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DEF A ∈ Mn(R) Hadamard matrix (1893): hij = ±1, columns orthogonal A ∈ Mn(±1) =⇒ (columns orthog ⇐⇒ rows orthog) ATA = nI ⇐⇒ A−1 = (1/n)AT ⇐⇒ AAT = nI

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DEF A ∈ Mn(R) Hadamard matrix (1893): hij = ±1, columns orthogonal A ∈ Mn(±1) =⇒ (columns orthog ⇐⇒ rows orthog) ATA = nI ⇐⇒ A−1 = (1/n)AT ⇐⇒ AAT = nI Example: SW1 =

• 1

1 1 −1

• CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

Hadamard matrices

DEF A ∈ Mn(R) Hadamard matrix (1893): hij = ±1, columns orthogonal A ∈ Mn(±1) =⇒ (columns orthog ⇐⇒ rows orthog) ATA = nI ⇐⇒ A−1 = (1/n)AT ⇐⇒ AAT = nI Example: SW1 =

• 1

1 1 −1

• 4 × 4 Sylvester–Walsh matrix (Sylvester 1867)

SW2 =             1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1             = SW1 ⊗ SW1

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DEF A ∈ Mn(R) Hadamard matrix (1893): hij = ±1, columns orthogonal A ∈ Mn(±1) =⇒ (columns orthog ⇐⇒ rows orthog) ATA = nI ⇐⇒ A−1 = (1/n)AT ⇐⇒ AAT = nI Example: SW1 =

• 1

1 1 −1

• 4 × 4 Sylvester–Walsh matrix (Sylvester 1867)

SW2 =             1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1             = SW1 ⊗ SW1 2k × 2k SWk = SW1 ⊗ SW1 ⊗ · · · ⊗ SW1

• k times

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DO A, B Hadamard =⇒ A ⊗ B Hadamard ∴ If ∃k × k, r × r Hadamard matrices =⇒ ∃kr × kr If ∃n × n Hadamard matrix, n ≥ 3, then 4 | n

• Proof. WLOG first row all ones (why?)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DO A, B Hadamard =⇒ A ⊗ B Hadamard ∴ If ∃k × k, r × r Hadamard matrices =⇒ ∃kr × kr If ∃n × n Hadamard matrix, n ≥ 3, then 4 | n

• Proof. WLOG first row all ones (why?)

row 1: 1 = (1, 1, 1, 1, . . . ) row i: x = (1, −1, −1, 1, . . . ) row j: y = (−1, 1, −1, 1, . . . ) 1 · x = 1 · y = x · y = 0 (x + 1)/2 ∈ Zn, (y + 1)/2 ∈ Zn ∴

x+1 2 · y+1 2

∈ Z

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

DO A, B Hadamard =⇒ A ⊗ B Hadamard ∴ If ∃k × k, r × r Hadamard matrices =⇒ ∃kr × kr If ∃n × n Hadamard matrix, n ≥ 3, then 4 | n

• Proof. WLOG first row all ones (why?)

row 1: 1 = (1, 1, 1, 1, . . . ) row i: x = (1, −1, −1, 1, . . . ) row j: y = (−1, 1, −1, 1, . . . ) 1 · x = 1 · y = x · y = 0 (x + 1)/2 ∈ Zn, (y + 1)/2 ∈ Zn ∴

x+1 2 · y+1 2

∈ Z but

x+1 2 · y+1 2

= 1

4( x · y

• + x · 1
• + y · 1
• + 1 · 1
• n

) = n

4

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} So far pretty sparse: powers of 2

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} So far pretty sparse: powers of 2 Raymond Paley (1933) q prime power, q ≡ −1 (mod 4) =⇒ ∃ (q + 1) × (q + 1) Hadamard matrix

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} So far pretty sparse: powers of 2 Raymond Paley (1933) q prime power, q ≡ −1 (mod 4) =⇒ ∃ (q + 1) × (q + 1) Hadamard matrix first q rows and columns indexed by Fq: for i ∈ Fq aii = −1 for i j ∈ Fq aij = 1 ⇐⇒ (∃x ∈ Fq)(x2 = i − j) add last row and column: all ones Example: p = 3             −1 −1 1 1 1 −1 −1 1 −1 1 −1 1 1 1 1 1            

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY? Prime Number Theorem: π(x) ∼ x ln x

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY? Prime Number Theorem: π(x) ∼ x ln x Jacques Hadamard and Charles de la Vallée Poussin

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY? Prime Number Theorem: π(x) ∼ x ln x Jacques Hadamard and Charles de la Vallée Poussin (1896)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY? Prime Number Theorem: π(x) ∼ x ln x Jacques Hadamard and Charles de la Vallée Poussin (1896) Prime Number Theorem in arithmetic progressions: If gcd(m, k) = 1 then π(x; m, k) := |{p ≤ x | p ≡ k (mod m)}| ∼ 1 ϕ(m) · x ln x

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

N = {n | ∃ Hadamard matrix of order n} Theorem (Paley 1933) q prime power, q ≡ −1 (mod 4) =⇒ q + 1 ∈ N ∴ density of N: |N ∩ [m]| m

• 1

2 ln m WHY? Prime Number Theorem: π(x) ∼ x ln x Jacques Hadamard and Charles de la Vallée Poussin (1896) Prime Number Theorem in arithmetic progressions: If gcd(m, k) = 1 then π(x; m, k) := |{p ≤ x | p ≡ k (mod m)}| ∼ 1 ϕ(m) · x ln x Charles de la Vallée Poussin

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

Paley also constructed Hadamard matrices of order 2(q + 1) for q a prime power, q ≡ 1 (mod 4) Paley’s constructions, together with Kronecker products, raise the density to |N ∩ [m]| m

• 1

(ln m)1−c for some small c (c < 1/4)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices

Paley also constructed Hadamard matrices of order 2(q + 1) for q a prime power, q ≡ 1 (mod 4) Paley’s constructions, together with Kronecker products, raise the density to |N ∩ [m]| m

• 1

(ln m)1−c for some small c (c < 1/4) If ∃n × n Hadamard matrix, n ≥ 3, then 4 | n CONJECTURE (Hadamard) 4 | n =⇒ ∃n × n Hadamard matrix Sylvester + Paley + Kronecker product verifies conjecture for n ≤ 88

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices N = {n | ∃ Hadamard matrix of order n} CONJECTURE (Hadamard) 4 | n =⇒ ∃n ∈ N Smallest open case: n = 668 = 4 · 167

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices N = {n | ∃ Hadamard matrix of order n} CONJECTURE (Hadamard) 4 | n =⇒ ∃n ∈ N Smallest open case: n = 668 = 4 · 167 next: 716 = 4 · 179

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices N = {n | ∃ Hadamard matrix of order n} CONJECTURE (Hadamard) 4 | n =⇒ ∃n ∈ N Smallest open case: n = 668 = 4 · 167 next: 716 = 4 · 179, 892 = 4 · 223, 1004 = 4 · 251, 1132 = 4 · 283, . . .

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Hadamard matrices N = {n | ∃ Hadamard matrix of order n} CONJECTURE (Hadamard) 4 | n =⇒ ∃n ∈ N Smallest open case: n = 668 = 4 · 167 next: 716 = 4 · 179, 892 = 4 · 223, 1004 = 4 · 251, 1132 = 4 · 283, . . . OPEN: lim sup

m→∞

|N ∩ m| m > 0 ?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

What is the value of the game? I.e., what is the fair price the Patron should pay to the House for the privilege of playing the game?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

P.C. Fishbum, N.J .A. Sloane

Plate 1. Berlekamp’s light-bulb game.

As to the exact values of R,, up to now it was only known that RI = 0, R2 = 1,

R3 =2,R,,

=4, R5=7, R+16, 22~R,~23, Rgs29 and32sRI,+37 [2,7]. The case FZ = 10 is of particular interest because of the existence at Bell Labs in Murray Hill of a game built by Elwyn Berlekamp some twenty years ago (see Plate 1). There are 100 light-bulbs, arranged in a 10 x 10 array. At the back of the box there are 100 switches, one for each bulb. On the front there are 20 switches, one for each row and column. Throwing one of the rear switches changes the state of a single bulb, while throwing one of the front switches complements a whole row 0~ coiumn of bulbs. For any initial set S of bulbs turned on using the rear switches, let f(S) be the minimal number of lights that can be achieved by throwing any combination of row and column switches. The problem, up to now unsolved, is to determine the maximum of f(S) over all choices of S.

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

Question 1: the payoff is ≥ 0 (Patron can always set the sum to be non-negative)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

Question 1: the payoff is ≥ 0 (Patron can always set the sum to be non-negative) Question 2: the payoff is ≥ n

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

Question 1: the payoff is ≥ 0 (Patron can always set the sum to be non-negative) Question 2: the payoff is ≥ n Question 3: the payoff is Ω(n3/2)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

Gale–Berlekamp switching game

The House has n2 switches fills n × n matrix with ±1 entries The Patron has 2n switches can change sign of rows and columns Payoff to Patron: n

i=1

n

j=1 aij

Question 1: the payoff is ≥ 0 (Patron can always set the sum to be non-negative) Question 2: the payoff is ≥ n Question 3: the payoff is Ω(n3/2) Question 4: the payoff is O(n3/2) (House can keep Patron from winning more)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics