The minimum number of disjoint pairs in set …page.mi.fu-berlin.de/shagnik/slides/13rsa.pdfThe...

The minimum number of disjoint pairs in setsystems and related problems

Shagnik Das

University of California, Los Angeles

Aug 7, 2013

Joint work with Wenying Gan and Benny Sudakov

Historical Background Our Results Concluding Remarks

Intersecting systems

Definition (Intersecting systems)

A set system F is said to be intersecting if F1 ∩ F2 6= ∅ for allF1,F2 ∈ F .

Observation

If a system F on [n] is intersecting, |F| ≤ 2n−1.

Proof: F can contain at most one of F ,F c for all F ⊂ [n].

Constructions:

Star: all sets containing 1(n odd) All sets of size at least n+1

2





Observation



Constructions:


2





Observation



Constructions:


2





Observation



Constructions:


2


Intersecting systems: Erdos-Ko-Rado

Theorem (Erdos-Ko-Rado, 1961)

Suppose n ≥ 2k, and F ⊂([n]k

)is intersecting. Then |F| ≤

(n−1k−1).

Extremal systems: stars

1

A star with centre 1






(n−1k−1).


1







(n−1k−1).


1



Beyond the threshold

Previous results answer the typical extremal problem

Question

How large can a structure be without containing a forbiddenconfiguration?

Gives rise to natural extension

Question

How many forbidden configurations must appear in largerstructures?


Beyond the threshold

Previous results answer the typical extremal problem

Question

How large can a structure be without containing a forbiddenconfiguration?

Gives rise to natural extension

Question

How many forbidden configurations must appear in largerstructures?


Beyond the threshold: one extra set

Warm-up: How many disjoint pairs must a system of 2n−1 + 1sets contain?

Answer: 1

Any maximal intersecting system can be extended to haveonly one disjoint pair

Let F be an intersecting system with 2n−1 sets, and letF0 ∈ F be minimal

Adding F c0 to F only creates the disjoint pair {F0,F c

0 }




Answer:

1




0 }




Answer: 1




0 }




Answer: 1




0 }




Answer: 1




0 }


Beyond the threshold: many extra sets

Previous argument:( [n]> n

2

)construction best to extend

Theorem (Frankl, 1977; Ahlswede, 1980)

Suppose∣∣∣( [n]>k

)∣∣∣ ≤ m ≤∣∣∣( [n]≥k)∣∣∣. Then the minimum number of

disjoint pairs in a system of m sets is attained by some F with( [n]>k

)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n




2






)⊂ F ⊂

( [n]≥k).

0 1

. . .

n−32

n−12

n+12

n+32

. . .

n − 1 n


The k-uniform setting

Extremal systems depend on pairs in([n]k

)

Question (Ahlswede, 1980)

Which systems in([n]k

)minimise the number of disjoint pairs?

Natural construction: union of stars

1 2 . . . s




)Question (Ahlswede, 1980)




1 2 . . . s








1

2 . . . s








1

2 . . . s








1

2 . . . s








1 2

. . . s








1 2

. . . s








1 2

. . . s








1 2 . . . s








1 2 . . . s








1 2 . . . s


A conjecture

Conjecture (Bollobas-Leader, 2000)

For small systems, a union of stars minimises the number ofdisjoint pairs.

A system is optimal iff its complement is

Conjecture ⇒ for large systems, a clique is optimal

Conjecture holds for k = 2

Theorem (Ahlswede-Katona, 1978)

The n-vertex graph with m edges and the minimal number ofdisjoint pairs of edges is:

a union of stars if m < 12

(n2

)− n

2 , and

a clique if m > 12

(n2

)+ n

2 .


A conjecture









(n2

)− n

2 , and

a clique if m > 12

(n2

)+ n

2 .


A conjecture









(n2

)− n

2 , and

a clique if m > 12

(n2

)+ n

2 .


A conjecture









(n2

)− n

2 , and

a clique if m > 12

(n2

)+ n

2 .


New result

We verify the Bollobas-Leader conjecture

Theorem (D.-Gan-Sudakov, 2013+)

Given n > 108k2s(k + s), and(nk

)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),

then the minimum number of disjoint pairs for a system of m setsin([n]k

)is attained by taking s − 1 full stars and a partial star.

Proof outline:

Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure


New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:



New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:



New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:

Step 1: Induction on s

Step 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure


New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:

Step 1: Induction on sStep 2: Existence of a popular element

Step 3: Existence of a cover of size sStep 4: Determining the exact structure


New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:

Step 1: Induction on sStep 2: Existence of a popular elementStep 3: Existence of a cover of size s

Step 4: Determining the exact structure


New result




)−(n−s+1

k

)≤ m ≤

(nk

)−(n−s

k

),



Proof outline:



Notation

dp(F) = # of disjoint pairs in Fdp(F ,G) = # of disjoint pairs between F and G

F(i) = {F ∈ F : i ∈ F}A(i) =

([n]k

)(i) = {F ⊂ [n] : |F | = k , i ∈ F}

X is a cover for F if for every F ∈ F , F ∩ X 6= ∅


Notation


F(i) = {F ∈ F : i ∈ F}A(i) =

([n]k

)(i) = {F ⊂ [n] : |F | = k , i ∈ F}



Notation


F(i) = {F ∈ F : i ∈ F}A(i) =

([n]k

)(i) = {F ⊂ [n] : |F | = k , i ∈ F}



Step one: induction

Base case: s = 1

Trivial: we have an intersecting system

Induction step: s ≥ 2

If F has a full star, say F(1), then we have

dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).

Induction ⇒ dp(F \ F(1)) minimised by a union of stars

Hence we may assume there are no full starsGiven F ∈ F and i ∈ [n], can replace F by a set containing i


Step one: induction

Base case: s = 1




dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).




Step one: induction

Base case: s = 1




dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).




Step one: induction

Base case: s = 1




dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).




Step one: induction

Base case: s = 1




dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).


Hence we may assume there are no full stars

Given F ∈ F and i ∈ [n], can replace F by a set containing i


Step one: induction

Base case: s = 1




dp(F) = dp(F(1),F \ F(1)) + dp(F \ F(1))

=

(n − k − 1

k − 1

)(m −

(n − 1

k − 1

))+ dp(F \ F(1)).




Step two: popular element

If F is extremal, we must have dp(F) ≤ 12

(1− 1

s

)m2

Goal: |F(i)| ≥ mks for some i (wlog i = 1)

Union-bound:

dp(F) =1

2

∑F∈F

dp(F ,F) ≥ 1

2

∑F∈F

(m −

∑i∈F|F(i)|

)

≥ 1

2

∑F∈F

(m − k |F(1)|) =1

2

(1− k |F(1)|

m

)m2.

Thus k|F(1)|m ≥ 1

s ⇒ |F(1)| ≥ mks




(1− 1

s

)m2


Union-bound:

dp(F) =1

2

∑F∈F

dp(F ,F) ≥ 1

2

∑F∈F

(m −

∑i∈F|F(i)|

)

≥ 1

2

∑F∈F

(m − k |F(1)|) =1

2

(1− k |F(1)|

m

)m2.

Thus k|F(1)|m ≥ 1

s ⇒ |F(1)| ≥ mks




(1− 1

s

)m2


Union-bound:

dp(F) =1

2

∑F∈F

dp(F ,F) ≥ 1

2

∑F∈F

(m −

∑i∈F|F(i)|

)

≥ 1

2

∑F∈F

(m − k |F(1)|) =1

2

(1− k |F(1)|

m

)m2.

Thus k|F(1)|m ≥ 1

s ⇒ |F(1)| ≥ mks




(1− 1

s

)m2


Union-bound:

dp(F) =1

2

∑F∈F

dp(F ,F) ≥ 1

2

∑F∈F

(m −

∑i∈F|F(i)|

)

≥ 1

2

∑F∈F

(m − k |F(1)|) =1

2

(1− k |F(1)|

m

)m2.

Thus k|F(1)|m ≥ 1

s ⇒ |F(1)| ≥ mks


Step three: small cover

We have dp(F ,F) = m − |∪i∈FF(i)|⇒ m −

∑i∈F |F(i)| ≤ dp(F ,F) ≤ m −maxi∈F |F(i)|

Since we can shift sets to F(1), we must have, for all F ∈ F ,∑i∈F |F(i)| ≥ |F(1)|

⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover

X is small:

km =∑

F∈F |F | =∑

i |F(i)| ≥∑

i∈X |F(i)| ≥ 1k |F(1)| |X |

⇒ |X | ≤ km|F(1)| ≤ k3s.






⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover

X is small:

km =∑

F∈F |F | =∑

i |F(i)| ≥∑

i∈X |F(i)| ≥ 1k |F(1)| |X |

⇒ |X | ≤ km|F(1)| ≤ k3s.






⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover

X is small:

km =∑

F∈F |F | =∑

i |F(i)| ≥∑

i∈X |F(i)| ≥ 1k |F(1)| |X |

⇒ |X | ≤ km|F(1)| ≤ k3s.






⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover

X is small:

km =∑

F∈F |F | =∑

i |F(i)| ≥∑

i∈X |F(i)| ≥ 1k |F(1)| |X |

⇒ |X | ≤ km|F(1)| ≤ k3s.






⇒ X = {i : |F(i)| ≥ 1k |F(1)|} is a cover

X is small:

km =∑

F∈F |F | =∑

i |F(i)| ≥∑

i∈X |F(i)| ≥ 1k |F(1)| |X |

⇒ |X | ≤ km|F(1)| ≤ k3s.


Step three: small cover (II)

Almost all intersections take place in X

X

F

Thus sets in F(i), i ∈ X , intersect (1 + o(1)) |F(i)| sets

⇒ for all i , j ∈ X , |F(i)| ≈ |F(j)| = (1 + o(1)) m|X |

⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s

)m2, we have |X | = s




X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s





X

F



⇒ dp(F) = 12

(1− 1

|X | + o(1))

m2.

Since dp(F) ≤ 12

(1− 1

s



Step four: exact structure

We may now assume [s] is a cover for F

Let A = ∪si=1A(i) be all sets meeting [s].

Then F ⊂ A; let G = A \ F .

Inclusion-exclusion: dp(F) = dp(A)− dp(G,A) + dp(G)

Minimised when F consists of s − 1 full stars and one partialstar

dp(G) = 0 as G is intersectingdp(G,A) =

∑G∈G dp(G ,A) maximised when |G ∩ X | = 1 for

all G ∈ G

�










all G ∈ G

�










all G ∈ G

�










all G ∈ G

�








dp(G) = 0 as G is intersecting

dp(G,A) =∑

G∈G dp(G ,A) maximised when |G ∩ X | = 1 forall G ∈ G

�










all G ∈ G

�










all G ∈ G

�


Further results

Characterisation of all extremal systems

t-disjoint pairs:

We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory

Can also minimise the number of q-matchings


Further results


t-disjoint pairs:

We say F1,F2 are t-disjoint if |F1 ∩ F2| < t

Using similar methods, we determine which small systemsminimise the number of t-disjoint pairsWhen t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory



Further results


t-disjoint pairs:

We say F1,F2 are t-disjoint if |F1 ∩ F2| < tUsing similar methods, we determine which small systemsminimise the number of t-disjoint pairs

When t = k − 1, this is known as the Kleitman-West problem,and arises in connection to information theory



Further results


t-disjoint pairs:




Further results


t-disjoint pairs:




Open problems

Minimising disjoint pairs:

Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions

Most probably intersecting (Katona-Katona-Katona):

Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting


Open problems


Union of stars for small systems, clique for large systems

What are the optimal systems in between?The full Bollobas-Leader conjecture provides a candidatefamily of constructions




Open problems


Union of stars for small systems, clique for large systemsWhat are the optimal systems in between?

The full Bollobas-Leader conjecture provides a candidatefamily of constructions




Open problems






Open problems






Open problems




Seek a set system whose random subsystems are mostprobably intersecting

Have previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting


Open problems




Seek a set system whose random subsystems are mostprobably intersectingHave previously seen close connections between the twoproblems (Russell-Walters)

More complicated relationship in the k-uniform setting


Open problems






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