Erd osRothschild for Intersecting Families Dennis Clemens 1 Shagnik - - PowerPoint PPT Presentation

Erd˝

• s–Rothschild for Intersecting Families

Dennis Clemens1 Shagnik Das2 Tuan Tran2

1Technische Universit¨

at Hamburg–Harburg

2Freie Universit¨

at Berlin

British Mathematical Colloquium, Bristol 23rd March 2016

Erd˝

• s–Rothschild for Intersecting Families

Dennis Clemens1 Shagnik Das2,3 Tuan Tran2

1Technische Universit¨

at Hamburg–Harburg

2Freie Universit¨

at Berlin

3(I’m this guy)

British Mathematical Colloquium, Bristol 23rd March 2016

Introduction A little mathematics Conclusion

{}: Outline

{} Introductory waffling {ER} The Erd˝

• s–Rothschild problem

{IF} Intersecting families {ER, IF} Erd˝

• s–Rothschild for intersecting families

Introduction A little mathematics Conclusion

{}: Outline

{} Introductory waffling {ER} The Erd˝

• s–Rothschild problem

{IF} Intersecting families {ER, IF} Erd˝

• s–Rothschild for intersecting families

Introduction A little mathematics Conclusion

{}: Outline

{} Introductory waffling {ER} The Erd˝

• s–Rothschild problem

{IF} Intersecting families {ER, IF} Erd˝

• s–Rothschild for intersecting families

Introduction A little mathematics Conclusion

{}: Outline

{} Introductory waffling {ER} The Erd˝

• s–Rothschild problem

{IF} Intersecting families {ER, IF} Erd˝

• s–Rothschild for intersecting families

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

The largest n-vertex triangle-free graph has

• n2/4
• edges.

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

ex(n, K3) =

• n2/4
• .

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

ex(n, K3) =

• n2/4
• .

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

ex(n, K3) =

• n2/4
• .

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

ex(n, K3) =

• n2/4
• .

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{}: Mantel’s Theorem

Theorem (Mantel, 1907)

ex(n, K3) =

• n2/4
• .

Mantel: tight

Extensions

Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Bipartite graph: any two-colouring works

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Bipartite graph: any two-colouring works

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Bipartite graph: any two-colouring works

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Extra edge: causes problems

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Extra edge: causes problems

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Extra edge: causes problems

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Extra edge: causes problems

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Extra edge: causes problems

Introduction A little mathematics Conclusion

{ER}: a new extension

Question

How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?

Conjecture (Erd˝

• s–Rothschild, 1974)

At most 2ex(n,K3).

Introduction A little mathematics Conclusion

{ER}: the known results

Theorem (Yuster, 1996)

If n is large enough, an n-vertex graph can have at most 2ex(n,K3) two-colourings without a monochromatic triangle.

Theorem (Alon–Balogh–Keevash–Sudakov, 2004)

Given r ∈ {2, 3}, k ≥ 2 and n large enough, an n-vertex graph can have at most rex(n,Kk) r-colourings without a monochromatic Kk.

Further results

◮ When r ≥ 4, maximum is greater than rex(n,Kk) ◮ Pikhurko–Yilma (2012): precise answer for r = 4,

k ∈ {3, 4} and n large

Introduction A little mathematics Conclusion

{IF}: the Erd˝

• s–Ko–Rado Theorem

Definition

A k-uniform set family F ⊆ [n]

k

• is said to be intersecting

if F1 ∩ F2 = ∅ for all F1, F2 ∈ F. 1

Star with centre 1

Theorem (Erd˝

• s–Ko–Rado, 1961)

If n ≥ 2k and F ⊆ [n]

k

• is intersecting,

|F| ≤ n − 1 k − 1

• .

Introduction A little mathematics Conclusion

{IF}: the Erd˝

• s–Ko–Rado Theorem

Definition

A k-uniform set family F ⊆ [n]

k

• is said to be intersecting

if F1 ∩ F2 = ∅ for all F1, F2 ∈ F. 1

Star with centre 1

Theorem (Erd˝

• s–Ko–Rado, 1961)

If n ≥ 2k and F ⊆ [n]

k

• is intersecting,

|F| ≤ n − 1 k − 1

• .

Introduction A little mathematics Conclusion

{IF}: the Erd˝

• s–Ko–Rado Theorem

Definition

A k-uniform set family F ⊆ [n]

k

• is said to be t-intersecting

if |F1 ∩ F2| ≥ t for all F1, F2 ∈ F. T

t-star with centre T

Theorem (Erd˝

• s–Ko–Rado, 1961;

Frankl, 1978; Wilson, 1984)

If n ≥ (t + 1)(k − t + 1) and F ⊆ [n]

k

• is t-intersecting,

|F| ≤ n − t k − t

• .

Introduction A little mathematics Conclusion

{ER, IF}: the Erd˝

• s–Rothschild extension

Definition

An (r, t)-colouring of a family F is an r-colouring of its members such that each colour class is t-intersecting.

Question

What is the maximum number

• f (r, t)-colourings

a family can have?

t-star: r(

n−t k−t)

Introduction A little mathematics Conclusion

{ER, IF}: the Erd˝

• s–Rothschild extension

Definition

An (r, t)-colouring of a family F is an r-colouring of its members such that each colour class is t-intersecting.

Question

What is the maximum number

• f (r, t)-colourings

a family can have?

t-star: r(

n−t k−t)

Introduction A little mathematics Conclusion

{ER, IF}: the Erd˝

• s–Rothschild extension

Definition

An (r, t)-colouring of a family F is an r-colouring of its members such that each colour class is t-intersecting.

Question

What is the maximum number

• f (r, t)-colourings

a family can have?

t-star: r(

n−t k−t)

Introduction A little mathematics Conclusion

{ER, IF}: the Erd˝

• s–Rothschild extension

Definition

An (r, t)-colouring of a family F is an r-colouring of its members such that each colour class is t-intersecting.

Question

What is the maximum number

• f (r, t)-colourings

a family can have?

t-star: r(

n−t k−t)

Introduction A little mathematics Conclusion

{ER, IF}: the Erd˝

• s–Rothschild extension

Definition

An (r, t)-colouring of a family F is an r-colouring of its members such that each colour class is t-intersecting.

Question

What is the maximum number

• f (r, t)-colourings

a family can have?

t-star: r(

n−t k−t)

Introduction A little mathematics Conclusion

{ER, IF}: Two-colourings are trivial

Observation

Let N0 be the size of the largest t-intersecting family. A family can have at most 2N0 (2, t)-colourings.

Proof.

Let F be a family, F0 ⊆ F a maximal t-intersecting subfamily. Fix some two-colouring of F0. For every F ∈ F \ F0, there must be F0 ∈ F0: |F ∩ F0| < t. Hence F and F0 cannot get the same colour. ⇒ at most one colour available for F. Thus F has at most 2|F0| ≤ 2N0 (2, t)-colourings.

Introduction A little mathematics Conclusion

{ER, IF}: Two-colourings are trivial

Observation

Let N0 be the size of the largest t-intersecting family. A family can have at most 2N0 (2, t)-colourings.

Proof.

Let F be a family, F0 ⊆ F a maximal t-intersecting subfamily. Fix some two-colouring of F0. For every F ∈ F \ F0, there must be F0 ∈ F0: |F ∩ F0| < t. Hence F and F0 cannot get the same colour. ⇒ at most one colour available for F. Thus F has at most 2|F0| ≤ 2N0 (2, t)-colourings.

Introduction A little mathematics Conclusion

{ER, IF}: Two-colourings are trivial

Observation

Let N0 be the size of the largest t-intersecting family. A family can have at most 2N0 (2, t)-colourings.

Proof.

Let F be a family, F0 ⊆ F a maximal t-intersecting subfamily. Fix some two-colouring of F0. For every F ∈ F \ F0, there must be F0 ∈ F0: |F ∩ F0| < t. Hence F and F0 cannot get the same colour. ⇒ at most one colour available for F. Thus F has at most 2|F0| ≤ 2N0 (2, t)-colourings.

Introduction A little mathematics Conclusion

{ER, IF}: Two-colourings are trivial

Observation

Let N0 be the size of the largest t-intersecting family. A family can have at most 2N0 (2, t)-colourings.

Proof.

Let F be a family, F0 ⊆ F a maximal t-intersecting subfamily. Fix some two-colouring of F0. For every F ∈ F \ F0, there must be F0 ∈ F0: |F ∩ F0| < t. Hence F and F0 cannot get the same colour. ⇒ at most one colour available for F. Thus F has at most 2|F0| ≤ 2N0 (2, t)-colourings.

Introduction A little mathematics Conclusion

{ER, IF}: Previous results

Setting Authors Parameters Hoppen k, t fixed Set families Kohayakawa r ≥ 2 fixed (2012) Lefmann n large Hoppen k, ℓ fixed ℓ-matchings Kohayakawa r ≥ 2 fixed (2015) Lefmann n large Hoppen k, q, t fixed Vector spaces Lefmann r ∈ {2, 3, 4} (2016+) Odermann n large

Introduction A little mathematics Conclusion

{ER, IF}: Previous results

Setting Authors Parameters Hoppen* k, t fixed Set families Kohayakawa r ≥ 2 fixed (2012) Lefmann+ n large Hoppen* k, ℓ fixed ℓ-matchings Kohayakawa r ≥ 2 fixed (2015) Lefmann+ n large Hoppen* k, q, t fixed Vector spaces Lefmann+ r ∈ {2, 3, 4} (2016+) Odermann n large

Introduction A little mathematics Conclusion

{ER, IF}: Our results – three colours

Simple unified proof

◮ Short proof that works in many settings

◮ Set families, matchings, vector spaces, permutations

◮ Common solution: “t-star” is extremal

Works for small n

◮ Method often gives very good bounds on parameters ◮ Set families:

◮ t = 1: require n ≥ (3 + o(1))k ◮ n > (t + 1)(k − t + 1) suffices for almost all t and k

◮ Vector spaces:

◮ t = 1: n > 2k suffices for almost all q and k

Introduction A little mathematics Conclusion

{ER, IF}: Our results – three colours

Simple unified proof

◮ Short proof that works in many settings

◮ Set families, matchings, vector spaces, permutations

◮ Common solution: “t-star” is extremal

Works for small n

◮ Method often gives very good bounds on parameters ◮ Set families:

◮ t = 1: require n ≥ (3 + o(1))k ◮ n > (t + 1)(k − t + 1) suffices for almost all t and k

◮ Vector spaces:

◮ t = 1: n > 2k suffices for almost all q and k

Introduction A little mathematics Conclusion

{ER, IF}: Our results – many colours

A rough structural result

◮ Can extend the method to the multicoloured case ◮ Common solution: union of r/3 t-stars ◮ Problem: not all unions are isomorphic

A more precise union

◮ Can use shifting arguments for finer structure ◮ Set families:

◮ r ≥ 5: recover Hoppen–Koyahakawa–Lefmann

◮ Vector spaces:

◮ 3|r: asymptotic results Containers Main theorem

Introduction A little mathematics Conclusion

{ER, IF}: Our results – many colours

A rough structural result

◮ Can extend the method to the multicoloured case ◮ Common solution: union of r/3 t-stars ◮ Problem: not all unions are isomorphic

A more precise union

◮ Can use shifting arguments for finer structure ◮ Set families:

◮ r ≥ 5: recover Hoppen–Koyahakawa–Lefmann

◮ Vector spaces:

◮ 3|r: asymptotic results Containers Main theorem

Introduction A little mathematics Conclusion

Containers: a brief introduction

Independent sets in Kneser graphs

Kneser graph: vertices [n]

k

• , edges F ∼ G ⇔ F ∩ G = ∅

Independent set in Kneser graph ↔ intersecting family What can we say about their distribution?

Containers

Container method (Balogh–Morris–Samotij, 2015; Saxton–Thomason, 2015):

◮ Small number of families Ci ◮ Each of size at most (1 + o(1))

n−1

k−1

• ◮ Every intersecting family contained in some Ci

Introduction A little mathematics Conclusion

Containers: a brief introduction

Independent sets in Kneser graphs

Kneser graph: vertices [n]

k

• , edges F ∼ G ⇔ F ∩ G = ∅

Independent set in Kneser graph ↔ intersecting family What can we say about their distribution?

Containers

Container method (Balogh–Morris–Samotij, 2015; Saxton–Thomason, 2015):

◮ Small number of families Ci ◮ Each of size at most (1 + o(1))

n−1

k−1

• ◮ Every intersecting family contained in some Ci

Introduction A little mathematics Conclusion

Containers: maximal families

Natural containers

Every intersecting family is contained in a maximal intersecting family Balogh–D.–Delcourt–Liu–Sharifzadeh (2015): very few of them

Advantages

Containers are intersecting

◮ Use existing extremal, stability results ◮ Container is either a star, or much smaller ◮ Intersections of containers are small

Much better numerical, structural control

Introduction A little mathematics Conclusion

Containers: maximal families

Natural containers

Every intersecting family is contained in a maximal intersecting family Balogh–D.–Delcourt–Liu–Sharifzadeh (2015): very few of them

Advantages

Containers are intersecting

◮ Use existing extremal, stability results ◮ Container is either a star, or much smaller ◮ Intersections of containers are small

Much better numerical, structural control

Introduction A little mathematics Conclusion

(3, t)-colourings: the main theorem

Notation

N0 Largest t-intersecting family N2 Maximum intersection of two maximal t-intersecting families M Number of maximal t-intersecting families

Theorem (Clemens–D.–Tran, 2016+)

Provided N0 − N2 − > 0, a family F can have at most 3N0 (3, t)-colourings, with equality if and only if F is a largest t-intersecting family.

Introduction A little mathematics Conclusion

(3, t)-colourings: the main theorem

Notation

N0 Largest t-intersecting family N2 Maximum intersection of two maximal t-intersecting families M Number of maximal t-intersecting families

Theorem (Clemens–D.–Tran, 2016+)

Provided N0 − N2 − 6 log2 M 2 log2 3 − 3 > 0, a family F can have at most 3N0 (3, t)-colourings, with equality if and only if F is a largest t-intersecting family.

Introduction A little mathematics Conclusion

(3, t)-colourings: the main theorem

Notation

N0 Largest t-intersecting family N2 Maximum intersection of two maximal t-intersecting families M Number of maximal t-intersecting families

Theorem (Clemens–D.–Tran, 2016+)

Provided N0 − N2 − 36 log2 M > 0, a family F can have at most 3N0 (3, t)-colourings, with equality if and only if F is a largest t-intersecting family.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof

Proof.

For every t-intersecting family I, fix some maximal M ⊇ I. Let F be a family with c(F) (3, t)-colourings. Each colouring: F = I1 ⊔ I2 ⊔ I3 → (M1, M2, M3). Pigeonhole: some (M1, M2, M3) is mapped to by c(M1, M2, M3) ≥ c(F)M−3 (3, t)-colourings. If M1 = M2 = M3 = M, then F ⊆ M ⇒ c(F) ≤ 3|M| ≤ 3N0. Equality: F = M, a largest t-intersecting family. Hence we may assume M1 = M2.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof

Proof.

For every t-intersecting family I, fix some maximal M ⊇ I. Let F be a family with c(F) (3, t)-colourings. Each colouring: F = I1 ⊔ I2 ⊔ I3 → (M1, M2, M3). Pigeonhole: some (M1, M2, M3) is mapped to by c(M1, M2, M3) ≥ c(F)M−3 (3, t)-colourings. If M1 = M2 = M3 = M, then F ⊆ M ⇒ c(F) ≤ 3|M| ≤ 3N0. Equality: F = M, a largest t-intersecting family. Hence we may assume M1 = M2.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof

Proof.

For every t-intersecting family I, fix some maximal M ⊇ I. Let F be a family with c(F) (3, t)-colourings. Each colouring: F = I1 ⊔ I2 ⊔ I3 → (M1, M2, M3). Pigeonhole: some (M1, M2, M3) is mapped to by c(M1, M2, M3) ≥ c(F)M−3 (3, t)-colourings. If M1 = M2 = M3 = M, then F ⊆ M ⇒ c(F) ≤ 3|M| ≤ 3N0. Equality: F = M, a largest t-intersecting family. Hence we may assume M1 = M2.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof

Proof.

For every t-intersecting family I, fix some maximal M ⊇ I. Let F be a family with c(F) (3, t)-colourings. Each colouring: F = I1 ⊔ I2 ⊔ I3 → (M1, M2, M3). Pigeonhole: some (M1, M2, M3) is mapped to by c(M1, M2, M3) ≥ c(F)M−3 (3, t)-colourings. If M1 = M2 = M3 = M, then F ⊆ M ⇒ c(F) ≤ 3|M| ≤ 3N0. Equality: F = M, a largest t-intersecting family. Hence we may assume M1 = M2.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof

Proof.

For every t-intersecting family I, fix some maximal M ⊇ I. Let F be a family with c(F) (3, t)-colourings. Each colouring: F = I1 ⊔ I2 ⊔ I3 → (M1, M2, M3). Pigeonhole: some (M1, M2, M3) is mapped to by c(M1, M2, M3) ≥ c(F)M−3 (3, t)-colourings. If M1 = M2 = M3 = M, then F ⊆ M ⇒ c(F) ≤ 3|M| ≤ 3N0. Equality: F = M, a largest t-intersecting family. Hence we may assume M1 = M2.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Bound c(M1, M2, M3) from above Let mi = # sets with i colours available c(M1, M2, M3) ≤ 2m23m3 m3 = |M1 ∩ M2 ∩ M3| ≤ N2 2m2 + 3m3 ≤ |M1| + |M2| + |M3| ≤ 3N0 ⇒ m2 ≤ 3

2 (N0 − m3)

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Bound c(M1, M2, M3) from above 1 2 1 2 1 2 3 Let mi = # sets with i colours available c(M1, M2, M3) ≤ 2m23m3 m3 = |M1 ∩ M2 ∩ M3| ≤ N2 2m2 + 3m3 ≤ |M1| + |M2| + |M3| ≤ 3N0 ⇒ m2 ≤ 3

2 (N0 − m3)

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Bound c(M1, M2, M3) from above 2 2 2 3 Let mi = # sets with i colours available c(M1, M2, M3) ≤ 2m23m3 m3 = |M1 ∩ M2 ∩ M3| ≤ N2 2m2 + 3m3 ≤ |M1| + |M2| + |M3| ≤ 3N0 ⇒ m2 ≤ 3

2 (N0 − m3)

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Bound c(M1, M2, M3) from above 2 2 2 3 Let mi = # sets with i colours available c(M1, M2, M3) ≤ 2m23m3 m3 = |M1 ∩ M2 ∩ M3| ≤ N2 2m2 + 3m3 ≤ |M1| + |M2| + |M3| ≤ 3N0 ⇒ m2 ≤ 3

2 (N0 − m3)

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Bound c(M1, M2, M3) from above 2 2 2 3 Let mi = # sets with i colours available c(M1, M2, M3) ≤ 2m23m3 m3 = |M1 ∩ M2 ∩ M3| ≤ N2 2m2 + 3m3 ≤ |M1| + |M2| + |M3| ≤ 3N0 ⇒ m2 ≤ 3

2 (N0 − m3)

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Hence c(M1, M2, M3) ≤ 2m23m3 ≤ 2

3 2 (N0−m3)3m3

= 2

3 2 N0

• 3 · 2− 3

2

m3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 , and c(F) ≤ c(M1, M2, M3)M3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 M3 = 3N0

• 2

3 2 · 3−1N0−N2− 6 log2 M 2 log2 3−3 < 3N0.

Thus, in this case, F cannot be extremal.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Hence c(M1, M2, M3) ≤ 2m23m3 ≤ 2

3 2 (N0−m3)3m3

= 2

3 2 N0

• 3 · 2− 3

2

m3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 , and c(F) ≤ c(M1, M2, M3)M3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 M3 = 3N0

• 2

3 2 · 3−1N0−N2− 6 log2 M 2 log2 3−3 < 3N0.

Thus, in this case, F cannot be extremal.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Hence c(M1, M2, M3) ≤ 2m23m3 ≤ 2

3 2 (N0−m3)3m3

= 2

3 2 N0

• 3 · 2− 3

2

m3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 , and c(F) ≤ c(M1, M2, M3)M3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 M3 = 3N0

• 2

3 2 · 3−1N0−N2− 6 log2 M 2 log2 3−3 < 3N0.

Thus, in this case, F cannot be extremal.

Introduction A little mathematics Conclusion

(3, t)-colourings: proof (cont’d)

Proof.

Hence c(M1, M2, M3) ≤ 2m23m3 ≤ 2

3 2 (N0−m3)3m3

= 2

3 2 N0

• 3 · 2− 3

2

m3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 , and c(F) ≤ c(M1, M2, M3)M3 ≤ 2

3 2 N0

• 3 · 2− 3

2

N2 M3 = 3N0

• 2

3 2 · 3−1N0−N2− 6 log2 M 2 log2 3−3 < 3N0.

Thus, in this case, F cannot be extremal.

Application Conclusion

Introduction A little mathematics Conclusion

Intersecting set families: parameters

N0: largest intersecting families

Erd˝

• s–Ko–Rado (1961): if n > 2k, largest intersecting families are

stars ⇒ N0 = n−1

k−1

• N2: intersection of two maximal intersecting families

Two distinct stars have intersection n−2

k−2

• Hilton–Milner (1967): largest non-star intersecting family

⇒ N2 = n−1

k−1

• −

n−k−1

k−1

• M: number of maximal families

Balogh–D.–Delcourt–Liu–Sharifzadeh (2015): maximal intersecting families determined by 2k−1

k−1

• sets

⇒ M ≤ n

k

(2k−1

k−1)

Introduction A little mathematics Conclusion

Intersecting set families: parameters

N0: largest intersecting families

Erd˝

• s–Ko–Rado (1961): if n > 2k, largest intersecting families are

stars ⇒ N0 = n−1

k−1

• N2: intersection of two maximal intersecting families

Two distinct stars have intersection n−2

k−2

• Hilton–Milner (1967): largest non-star intersecting family

⇒ N2 = n−1

k−1

• −

n−k−1

k−1

• M: number of maximal families

Balogh–D.–Delcourt–Liu–Sharifzadeh (2015): maximal intersecting families determined by 2k−1

k−1

• sets

⇒ M ≤ n

k

(2k−1

k−1)

Introduction A little mathematics Conclusion

Intersecting set families: parameters

N0: largest intersecting families

Erd˝

• s–Ko–Rado (1961): if n > 2k, largest intersecting families are

stars ⇒ N0 = n−1

k−1

• N2: intersection of two maximal intersecting families

Two distinct stars have intersection n−2

k−2

• Hilton–Milner (1967): largest non-star intersecting family

⇒ N2 = n−1

k−1

• −

n−k−1

k−1

• M: number of maximal families

Balogh–D.–Delcourt–Liu–Sharifzadeh (2015): maximal intersecting families determined by 2k−1

k−1

• sets

⇒ M ≤ n

k

(2k−1

k−1)

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show N0 − N2 − 36 log2 M > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show N0 − N2 − 36 log2 M > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show n − k − 1 k − 1

• − 36

2k − 1 k − 1

• log2

n k

• > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show n − k − 1 k − 1

• − 36

2k − 1 k − 1

• k log2

ne k

• > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show n − k − 1 k − 1

• − 36

2k − 1 k − 1

• k log2

ne k

• > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show 2k + O(log k) k − 1

• − 36

2k − 1 k − 1

• k log2 (4e) > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show 2O(log k) 2k − 1 k − 1

• − 36

2k − 1 k − 1

• k log2 (4e) > 0,

Introduction A little mathematics Conclusion

Intersecting set families: applying the theorem

Corollary (Clemens–D.–Tran, 2016+)

If n ≥ 3k + O (log k), a family F ⊆ [n]

k

• can have at most 3(n−1

k−1)

(3, 1)-colourings, with equality if and only if F is a star.

Proof.

Suffices to show kO(1) 2k − 1 k − 1

• − 36

2k − 1 k − 1

• k log2 (4e) > 0,

and this is true!

Introduction A little mathematics Conclusion

Open problems

Better bounds

◮ For (3, 1)-colourings of set families, we needed

n ≥ 3k + O(log k)

◮ Are stars still extremal for n ≥ (2 + o(1))k? ◮ For (r, 1)-colourings, obtain results for n ≥ ck2 ◮ Is a union of ⌈r/3⌉ stars extremal when n = O(k)?

Other settings

◮ Non-uniform intersecting families? ◮ Two-colour triviality: at most 22n−1 (2, 1)-colourings ◮ Conjecture:

• [n]

≥⌊n/2⌋

• has most (3, 1)-colourings

Introduction A little mathematics Conclusion

Open problems

Better bounds

◮ For (3, 1)-colourings of set families, we needed

n ≥ 3k + O(log k)

◮ Are stars still extremal for n ≥ (2 + o(1))k? ◮ For (r, 1)-colourings, obtain results for n ≥ ck2 ◮ Is a union of ⌈r/3⌉ stars extremal when n = O(k)?

Other settings

◮ Non-uniform intersecting families? ◮ Two-colour triviality: at most 22n−1 (2, 1)-colourings ◮ Conjecture:

• [n]

≥⌊n/2⌋

• has most (3, 1)-colourings

Introduction A little mathematics Conclusion