Removal Lemma for Nearly-Intersecting Families Shagnik Das Freie - - PowerPoint PPT Presentation

Removal Lemma for Nearly-Intersecting Families

Shagnik Das

Freie Universit¨ at, Berlin, Germany

August 25, 2015 Joint work with Tuan Tran

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Intersecting families

Definition (Intersecting families) A family of sets F is intersecting if F1 ∩ F2 = ∅ for all F1, F2 ∈ F.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Intersecting families

Definition (Intersecting families) A family of sets F is intersecting if F1 ∩ F2 = ∅ for all F1, F2 ∈ F. Notation [n] = {1, 2, . . . , n} - ground set for our set families [n]

k

• = {F ⊂ [n] : |F| = k}

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Intersecting families

Definition (Intersecting families) A family of sets F is intersecting if F1 ∩ F2 = ∅ for all F1, F2 ∈ F. Notation [n] = {1, 2, . . . , n} - ground set for our set families [n]

k

• = {F ⊂ [n] : |F| = k}

dp(F) = |{F, G ∈ F : F ∩ G = ∅}| F intersecting ⇔ dp(F) = 0

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Erd˝

• s–Ko–Rado theorem

Theorem (Erd˝

• s–Ko–Rado, 1961)

If k ≤ 1

2n, and F ⊆

[n]

k

• is intersecting, then |F| ≤

n−1

k−1

• .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Erd˝

• s–Ko–Rado theorem

Theorem (Erd˝

• s–Ko–Rado, 1961)

If k ≤ 1

2n, and F ⊆

[n]

k

• is intersecting, then |F| ≤

n−1

k−1

• .

If k > 1

2n,

[n]

k

• itself is intersecting

If k < 1

2n, unique extremal families are stars: all sets

containing some fixed element i ∈ [n]

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Erd˝

• s–Ko–Rado theorem

Theorem (Erd˝

• s–Ko–Rado, 1961)

If k ≤ 1

2n, and F ⊆

[n]

k

• is intersecting, then |F| ≤

n−1

k−1

• .

If k > 1

2n,

[n]

k

• itself is intersecting

If k < 1

2n, unique extremal families are stars: all sets

containing some fixed element i ∈ [n] 1

A star with centre 1

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Stability

Question (Stability) What can we say about the structure of large intersecting families?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Stability

Question (Stability) What can we say about the structure of large intersecting families? Theorem (Hilton–Milner, 1967) If k < 1

2n, and F ⊆

[n]

k

• is intersecting with

|F| > n−1

k−1

• −

n−k−1

k−1

• + 1, then F is contained in a star.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Stability

Question (Stability) What can we say about the structure of large intersecting families? Theorem (Hilton–Milner, 1967) If k < 1

2n, and F ⊆

[n]

k

• is intersecting with

|F| > n−1

k−1

• −

n−k−1

k−1

• + 1, then F is contained in a star.

Bound is best-possible, but . . . . . . the Hilton–Milner families have all but one set in a star.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Robust stability

Theorem (Friedgut, 2008) Let ζ > 0, and let ζn ≤ k ≤ 1

2 − ζ

• n. There is some c = c(ζ)

such that for every intersecting F ⊆ [n]

k

• with |F| ≥ (1 − ε)

n−1

k−1

• there is a star S with |F \ S| ≤ cε

n−1

k−1

• .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Robust stability

Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) Let ζ > 0, and let 2 ≤ k ≤ 1

2 − ζ

• n. There is some c = c(ζ)

such that for every intersecting F ⊆ [n]

k

• with |F| ≥ (1 − ε)

n−1

k−1

• there is a star S with |F \ S| ≤ cε

n−1

k−1

• .

Much stronger result obtained when k = o(n)

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Robust stability

Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) Let ζ > 0, and let 2 ≤ k ≤ 1

2 − ζ

• n. There is some c = c(ζ)

such that for every intersecting F ⊆ [n]

k

• with |F| ≥ (1 − ε)

n−1

k−1

• there is a star S with |F \ S| ≤ cε

n−1

k−1

• .

Much stronger result obtained when k = o(n) Theorem (Keevash–Mubayi, 2010) For every ε > 0 there is a δ > 0 such that for n sufficiently large and εn ≤ k ≤ 1

2n − 1, if F ⊆

[n]

k

• is intersecting with

|F| ≥

• 1 − δ · n−2k

n

n−1

k−1

• , then there is some star S with

|F \ S| ≤ ε n−1

k−1

• .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Robust stability

Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) Let ζ > 0, and let 2 ≤ k ≤ 1

2 − ζ

• n. There is some c = c(ζ)

such that for every intersecting F ⊆ [n]

k

• with |F| ≥ (1 − ε)

n−1

k−1

• there is a star S with |F \ S| ≤ cε

n−1

k−1

• .

Much stronger result obtained when k = o(n) Theorem (Keevash–Mubayi, 2010) For every ε > 0 there is a δ > 0 such that for n sufficiently large and εn ≤ k ≤ 1

2n − 1, if F ⊆

[n]

k

• is intersecting with

|F| ≥

• 1 − δ · n−2k

n

n−1

k−1

• , then there is some star S with

|F \ S| ≤ ε n−1

k−1

• .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Nearly-intersecting families

Previous results: large intersecting families are close to stars

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Nearly-intersecting families

Previous results: large intersecting families are close to stars Recent directions require study of nearly-intersecting families:

Families with relatively few disjoint pairs Useful for studying supersaturation, probabilistic versions

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Nearly-intersecting families

Previous results: large intersecting families are close to stars Recent directions require study of nearly-intersecting families:

Families with relatively few disjoint pairs Useful for studying supersaturation, probabilistic versions

Question What can we say about the structure of set families with few disjoint pairs?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Removal lemma

Theorem (Friedgut–Regev) Let ζ > 0, and ζn ≤ k ≤ 1

2 − ζ

• n. If F ⊆

[n]

k

• is such that

dp(F) is small, then F can be made intersecting by removing few sets.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Removal lemma

Theorem (Friedgut–Regev) Let ζ > 0, and ζn ≤ k ≤ 1

2 − ζ

• n. If F ⊆

[n]

k

• is such that

dp(F) is small, then F can be made intersecting by removing few sets. “Few disjoint pairs ⇒ ε-close to intersecting”

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Removal lemma

Theorem (Friedgut–Regev) Let ζ > 0, and ζn ≤ k ≤ 1

2 − ζ

• n. For every ε > 0 there is a

δ > 0 such that if F ⊆ [n]

k

• has dp(F) ≤ δ |F|

n−k

k

• , then F can

be made intersecting by removing at most ε n

k

• sets.

“Few disjoint pairs ⇒ ε-close to intersecting”

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Removal lemma

Theorem (Friedgut–Regev) Let ζ > 0, and ζn ≤ k ≤ 1

2 − ζ

• n. For every ε > 0 there is a

δ > 0 such that if F ⊆ [n]

k

• has dp(F) ≤ δ |F|

n−k

k

• , then F can

be made intersecting by removing at most ε n

k

• sets.

“Few disjoint pairs ⇒ ε-close to intersecting” Works for any F, regardless of closest intersecting family

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our removal lemma

Theorem (D.–Tran) If 2 ≤ k < 1

2n, then for every F ⊂

[n]

k

• with

|F| close to n − 1 k − 1

• and dp(F) small,

there is some star S such that |F∆S| is small.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our removal lemma

Theorem (D.–Tran) If 2 ≤ k < 1

2n, then for every F ⊂

[n]

k

• with

|F| close to n − 1 k − 1

• and dp(F) small,

there is some star S such that |F∆S| is small. “Large nearly-intersecting families are close to stars”

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our removal lemma

Theorem (D.–Tran) If 2 ≤ k < 1

2n, then for every F ⊂

[n]

k

• with

|F| close to n − 1 k − 1

• and dp(F) small,

there is some star S such that |F∆S| is small. “Large nearly-intersecting families are close to stars” Works for all k, but only for large families

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our removal lemma

Theorem (D.–Tran) There are constants c, C > 0 such that if 2 ≤ k < 1

2n and

max{|α| , β} ≤ c · n−2k

n

, then for every F ⊂ [n]

k

• with

|F| = (1 − α) n − 1 k − 1

• and dp(F) ≤ β

n − 1 k − 1 n − k − 1 k − 1

• ,

there is some star S such that |F∆S| ≤ C(α + 2β)

n n−2k

n−1

k−1

• .

“Large nearly-intersecting families are close to stars” Works for all k, but only for large families Provides sharp quantitative bounds

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Kneser graphs

Definition (Kneser graphs) The Kneser graph KG(n, k) is a graph with vertices [n]

k

• and an

edge between two sets if and only if they are disjoint.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Kneser graphs

Definition (Kneser graphs) The Kneser graph KG(n, k) is a graph with vertices [n]

k

• and an

edge between two sets if and only if they are disjoint. Independent sets ↔ intersecting families

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Kneser graphs

Definition (Kneser graphs) The Kneser graph KG(n, k) is a graph with vertices [n]

k

• and an

edge between two sets if and only if they are disjoint. Independent sets ↔ intersecting families Theorem (Erd˝

• s–Ko–Rado, 1961)

For k ≤ 1

2n, α (KG(n, k)) =

n−1

k−1

• .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Random Kneser subgraphs

Definition (Random Kneser subgraphs) For 0 ≤ p ≤ 1, let KGp(n, k) denote the random (spanning) subgraph of KG(n, k) obtained by retaining each edge independently with probability p.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Random Kneser subgraphs

Definition (Random Kneser subgraphs) For 0 ≤ p ≤ 1, let KGp(n, k) denote the random (spanning) subgraph of KG(n, k) obtained by retaining each edge independently with probability p. KGp(n, k) ⊆ KG(n, k) ⇒ α(KGp(n, k)) ≥ n−1

k−1

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Random Kneser subgraphs

Definition (Random Kneser subgraphs) For 0 ≤ p ≤ 1, let KGp(n, k) denote the random (spanning) subgraph of KG(n, k) obtained by retaining each edge independently with probability p. KGp(n, k) ⊆ KG(n, k) ⇒ α(KGp(n, k)) ≥ n−1

k−1

• Question (Sparse Erd˝
• s–Ko–Rado)

For which p do we have α(KGp(n, k)) = n−1

k−1

• ?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Random Kneser subgraphs

Definition (Random Kneser subgraphs) For 0 ≤ p ≤ 1, let KGp(n, k) denote the random (spanning) subgraph of KG(n, k) obtained by retaining each edge independently with probability p. KGp(n, k) ⊆ KG(n, k) ⇒ α(KGp(n, k)) ≥ n−1

k−1

• Question (Sparse Erd˝
• s–Ko–Rado)

For which p do we have sparse EKR?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Simulation: KG(5, 2)

12 12 12 34 34 34 34 15 15 15 23 23 45 45 45 45 35 35 25 25 24 24 24 24 14 14 14 14 13 13 13 1 p p p p p

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Superstars

Given star S, set F / ∈ S: n−k−1

k−1

• edges from F to S

⇒ P(S ∪ {F} independent) = (1 − p)(n−k−1

k−1 )

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Superstars

Given star S, set F / ∈ S: n−k−1

k−1

• edges from F to S

⇒ P(S ∪ {F} independent) = (1 − p)(n−k−1

k−1 )

n stars S, n−1

k

• sets F /

∈ S ⇒ n n−1

k

• possible superstars

⇒ threshold pc :=

log(n(n−1

k ))

(n−k−1

k−1 )

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Superstars

Given star S, set F / ∈ S: n−k−1

k−1

• edges from F to S

⇒ P(S ∪ {F} independent) = (1 − p)(n−k−1

k−1 )

n stars S, n−1

k

• sets F /

∈ S ⇒ n n−1

k

• possible superstars

⇒ threshold pc :=

log(n(n−1

k ))

(n−k−1

k−1 )

p < (1−ε)pc ⇒ w.h.p. there is a superstar ⇒ no sparse EKR

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Superstars

Given star S, set F / ∈ S: n−k−1

k−1

• edges from F to S

⇒ P(S ∪ {F} independent) = (1 − p)(n−k−1

k−1 )

n stars S, n−1

k

• sets F /

∈ S ⇒ n n−1

k

• possible superstars

⇒ threshold pc :=

log(n(n−1

k ))

(n−k−1

k−1 )

p < (1−ε)pc ⇒ w.h.p. there is a superstar ⇒ no sparse EKR p > (1 + ε)pc ⇒ w.h.p. there are no superstars

Can there be other large independent sets?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Previous results

Theorem (Bollob´ as–Narayanan–Raigorodskii, 2014+) Fix ε > 0, and let 2 ≤ k = o(n1/3). If p ≥ (1 + ε)pc, P( sparse EKR ) → 1 as n → ∞. Moreover, with high probability the only maximum independent sets are stars.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Previous results

Theorem (Bollob´ as–Narayanan–Raigorodskii, 2014+) Fix ε > 0, and let 2 ≤ k = o(n1/3). If p ≥ (1 + ε)pc, P( sparse EKR ) → 1 as n → ∞. Moreover, with high probability the only maximum independent sets are stars. Theorem (Balogh–Bollob´ as–Narayanan, 2014+) For every ζ > 0, there is a constant c = c(ζ) > 0 such that if k ≤ 1

2 − ζ

• n, then, as n → ∞,

P( sparse EKR ) → 1 if p ≥ n − 1 k − 1 −c .

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our result

Theorem (D.–Tran) There are constants c, C > 0 such that P( sparse EKR ) → 1 if either of the following hold:

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our result

Theorem (D.–Tran) There are constants c, C > 0 such that P( sparse EKR ) → 1 if either of the following hold: (i) k ≤ cn and p ≥ (1 + ε)pc, for any ε = ω(k−1), or

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our result

Theorem (D.–Tran) There are constants c, C > 0 such that P( sparse EKR ) → 1 if either of the following hold: (i) k ≤ cn and p ≥ (1 + ε)pc, for any ε = ω(k−1), or (ii) k ≤ 1

2(n − 3) and p ≥ C · n n−2k · pc.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Our result

Theorem (D.–Tran) There are constants c, C > 0 such that P( sparse EKR ) → 1 if either of the following hold: (i) k ≤ cn and p ≥ (1 + ε)pc, for any ε = ω(k−1), or (ii) k ≤ 1

2(n − 3) and p ≥ C · n n−2k · pc.

Devlin and Kahn prove threshold results for k ∼ 1

2n

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Proof

Need to check there are no non-star independent sets F of size n−1

k−1

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Proof

Need to check there are no non-star independent sets F of size n−1

k−1

• Simple union bound:

P (large non-star independent set) ≤

• F

(1 − p)dp(F).

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Proof

Need to check there are no non-star independent sets F of size n−1

k−1

• Simple union bound:

P (large non-star independent set) ≤

• F

(1 − p)dp(F). If dp(F) is large, contribution is insignificant

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Proof

Need to check there are no non-star independent sets F of size n−1

k−1

• Simple union bound:

P (large non-star independent set) ≤

• F

(1 − p)dp(F). If dp(F) is large, contribution is insignificant If dp(F) is small: Removal lemma ⇒ F is close to a star ⇒ few such families ⇒ total contribution is insignificant

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Proof

Need to check there are no non-star independent sets F of size n−1

k−1

• Simple union bound:

P (large non-star independent set) ≤

• F

(1 − p)dp(F). If dp(F) is large, contribution is insignificant If dp(F) is small: Removal lemma ⇒ F is close to a star ⇒ few such families ⇒ total contribution is insignificant

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Statement

Theorem (D.–Tran) If 2 ≤ k < 1

2n, then for every F ⊂

[n]

k

• with

|F| close to n − 1 k − 1

• and dp(F) small,

there is some star S such that |F∆S| is small.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Spectral framework

Consider functions f : [n]

k

• → R as f (x1, x2, . . . , xn) defined
• n {x :

i xi = k} or as vectors (f (F))F∈([n]

k )

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Spectral framework

Consider functions f : [n]

k

• → R as f (x1, x2, . . . , xn) defined
• n {x :

i xi = k} or as vectors (f (F))F∈([n]

k )

Fact If f is the characteristic function/vector of F ⊆ [n]

k

• , and A is the

adjacency matrix of the Kneser graph KG(n, k), then f TAf = 2dp(F).

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Spectral framework

Consider functions f : [n]

k

• → R as f (x1, x2, . . . , xn) defined
• n {x :

i xi = k} or as vectors (f (F))F∈([n]

k )

Fact If f is the characteristic function/vector of F ⊆ [n]

k

• , and A is the

adjacency matrix of the Kneser graph KG(n, k), then f TAf = 2dp(F). Definition (Affine functions) A function f : [n]

k

• → R is affine if f (x) = a0 + n

i=1 aixi for some

coefficients a0, a1, . . . , an.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Lemma 1

Lemma Suppose 2 ≤ k < 1

2n, and F ⊆

[n]

k

• is a family with |F| close to

n−1

k−1

• and dp(F) small. If f :

[n]

k

• → {0, 1} is the characteristic

function of F, there is some affine function g : [n]

k

• → R with

f − g2 small.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Lemma 1

Lemma Suppose 2 ≤ k < 1

2n, and F ⊆

[n]

k

• is a family with |F| close to

n−1

k−1

• and dp(F) small. If f :

[n]

k

• → {0, 1} is the characteristic

function of F, there is some affine function g : [n]

k

• → R with

f − g2 small. Lov´ asz (1979): determined spectrum of Kneser graphs Eigenspaces corresponding to two most significant eigenvalues are precisely the affine functions

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Lemma 2

Lemma (Filmus, 2014+) There is some constant C > 0, such that if 2 ≤ k ≤ 1

2n and

ε <

k 128n, and f :

[n]

k

• → {0, 1} is ε-close to an affine function,

then there is some S ⊂ [n] of size |S| ≤ t = max

• 1, Cn√ε

k

• such

that either f or 1 − f is (Cε)-close to maxi∈S xi.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Lemma 2

Lemma (Filmus, 2014+) There is some constant C > 0, such that if 2 ≤ k ≤ 1

2n and

ε <

k 128n, and f :

[n]

k

• → {0, 1} is ε-close to an affine function,

then there is some S ⊂ [n] of size |S| ≤ t = max

• 1, Cn√ε

k

• such

that either f or 1 − f is (Cε)-close to maxi∈S xi. “Boolean + close to affine ⇒ almost determined by few variables” Version of the Friedgut–Kalai–Naor theorem for these uniform slices of the Boolean cube

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Lemma 2

Lemma (Filmus, 2014+) There is some constant C > 0, such that if 2 ≤ k ≤ 1

2n and

ε <

k 128n, and f :

[n]

k

• → {0, 1} is ε-close to an affine function,

then there is some S ⊂ [n] of size |S| ≤ t = max

• 1, Cn√ε

k

• such

that either f or 1 − f is (Cε)-close to maxi∈S xi. “Boolean + close to affine ⇒ almost determined by few variables” Version of the Friedgut–Kalai–Naor theorem for these uniform slices of the Boolean cube Gs = [n]

k

• \

[n]\[s]

k

• : characteristic function gs = maxi∈[s] xi.

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Counting

Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Counting

Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t G0 G0 G2 Gt G1 G1

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Counting

Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t G0 G0 G2 Gt G1 G1 Too small Too big Too big Too big dp too big Just right!

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Counting

Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t G0 G0 G2 Gt G1 G1 Too small Too big Too big Too big dp too big Just right!

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Counting

Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t G0 G0 G2 Gt G1 G1 Too small Too big Too big Too big dp too big Just right! But G1 is precisely a star

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Open questions

Sharp threshold for sparse Erd˝

• s–Ko–Rado theorem when

k ∼ 1

2n: is p ≥ (1 + ε)pc always sufficient?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Open questions

Sharp threshold for sparse Erd˝

• s–Ko–Rado theorem when

k ∼ 1

2n: is p ≥ (1 + ε)pc always sufficient?

Hitting time: if we randomly remove edges from the Kneser graph, does α(G) > n−1

k−1

• ccur precisely when a superstar is

born?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion

Open questions

Sharp threshold for sparse Erd˝

• s–Ko–Rado theorem when

k ∼ 1

2n: is p ≥ (1 + ε)pc always sufficient?

Hitting time: if we randomly remove edges from the Kneser graph, does α(G) > n−1

k−1

• ccur precisely when a superstar is

born? Other applications of intersecting removal lemmas?

Erd˝

• s–Ko–Rado

Removal Lemmas Sparse EKR Removal Proof Conclusion