Post on 06-Apr-2020
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
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
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
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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?
Historical Background Our Results Concluding Remarks
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?
Historical Background Our Results Concluding Remarks
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 }
Historical Background Our Results Concluding Remarks
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 }
Historical Background Our Results Concluding Remarks
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 }
Historical Background Our Results Concluding Remarks
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 }
Historical Background Our Results Concluding Remarks
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 }
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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 .
Historical Background Our Results Concluding Remarks
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 .
Historical Background Our Results Concluding Remarks
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 .
Historical Background Our Results Concluding Remarks
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 .
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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 s
Step 2: Existence of a popular elementStep 3: Existence of a cover of size sStep 4: Determining the exact structure
Historical Background Our Results Concluding Remarks
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 element
Step 3: Existence of a cover of size sStep 4: Determining the exact structure
Historical Background Our Results Concluding Remarks
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 s
Step 4: Determining the exact structure
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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= ∅
Historical Background Our Results Concluding Remarks
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= ∅
Historical Background Our Results Concluding Remarks
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= ∅
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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 stars
Given F ∈ F and i ∈ [n], can replace F by a set containing i
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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.
Historical Background Our Results Concluding Remarks
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.
Historical Background Our Results Concluding Remarks
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.
Historical Background Our Results Concluding Remarks
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.
Historical Background Our Results Concluding Remarks
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.
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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 intersecting
dp(G,A) =∑
G∈G dp(G ,A) maximised when |G ∩ X | = 1 forall G ∈ G
�
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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
�
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
Further results
Characterisation of all extremal systems
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
Can also minimise the number of q-matchings
Historical Background Our Results Concluding Remarks
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 pairs
When 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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
Open problems
Minimising disjoint pairs:
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
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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 intersecting
Have previously seen close connections between the twoproblems (Russell-Walters)More complicated relationship in the k-uniform setting
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks
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
Historical Background Our Results Concluding Remarks