Removal Lemma for Nearly-Intersecting Families
Shagnik Das
Freie Universität, Berlin, Germany
August 11, 2015
Joint work with Tuan Tran
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Intersecting families
Definition (Intersecting families)
A family of sets F is intersecting if F1 ∩ F2 6= ∅ for all F1 , F2 ∈ F.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Intersecting families
Definition (Intersecting families)
A family of sets F is intersecting if F1 ∩ F2 6= ∅ for all F1 , F2 ∈ F.
Notation
[n] = {1, 2, . . . , n} - ground set for our set families
[n]
k = {F ⊂ [n] : |F | = k}
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Intersecting families
Definition (Intersecting families)
A family of sets F is intersecting if F1 ∩ F2 6= ∅ 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
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Erdős–Ko–Rado theorem
Theorem (Erdős–Ko–Rado, 1961)
If k ≤ 12 n, and F ⊆ [n]
k is intersecting, then |F| ≤
n−1
k−1
.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Erdős–Ko–Rado theorem
Theorem (Erdős–Ko–Rado, 1961)
If k ≤ 12 n, and F ⊆ [n]
k is intersecting, then |F| ≤
n−1
k−1
Remarks
If k > 12 n, [n]
k itself is intersecting
If k < 12 n, unique extremal families are stars: all sets
containing some fixed element i ∈ [n]
.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Erdős–Ko–Rado theorem
Theorem (Erdős–Ko–Rado, 1961)
If k ≤ 12 n, and F ⊆ [n]
k is intersecting, then |F| ≤
n−1
k−1
Remarks
If k > 12 n, [n]
k itself is intersecting
If k < 12 n, unique extremal families are stars: all sets
containing some fixed element i ∈ [n]
1
A star with centre 1
.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Stability
Question (Stability)
What can we say about the structure of large intersecting families?
Erdős–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 < 12 n, and F ⊆ [n]
k is intersecting with
n−1
|F| > k−1
− n−k−1
+ 1, then F is contained in a star.
k−1
Erdős–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 < 12 n, and F ⊆ [n]
k is intersecting with
n−1
|F| > k−1
− n−k−1
+ 1, then F is contained in a star.
k−1
Remarks
Bound is best-possible, but . . .
. . . the Hilton–Milner families have all but one set in a star.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Robust stability
Theorem (Friedgut, 2008; Dinur–Friedgut, 2009)
Let ζ > 0, and let 2 ≤ k ≤ 12 − ζ n. There is some c = c(ζ)
n−1
such that for every intersecting F ⊆ [n]
k with |F| ≥ (1 − ε) k−1
n−1
there is a star S with |F \ S| ≤ cε k−1
.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Robust stability
Theorem (Friedgut, 2008; Dinur–Friedgut, 2009)
Let ζ > 0, and let 2 ≤ k ≤ 12 − ζ n. There is some c = c(ζ)
n−1
such that for every intersecting F ⊆ [n]
k with |F| ≥ (1 − ε) k−1
n−1
there is a star S with |F \ S| ≤ cε k−1
.
Theorem (Keevash–Mubayi, 2010)
For every ε > 0 there is a δ > 0 such
that for n sufficiently large
[n]
1
and εn ≤ k ≤ 2 n − 1, if F ⊆ k is intersecting with
n−1
|F| ≥ 1 − δ · n−2k
n
k−1 , then there is some star S with
n−1
|F \ S| ≤ ε k−1 .
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Nearly-intersecting families
Previous results: large intersecting families are close to stars
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Nearly-intersecting families
Previous results: large intersecting families are close to stars
Recent trends require study of nearly-intersecting families:
Families with relatively few disjoint pairs
Useful for studying supersaturation, probabilistic versions
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Nearly-intersecting families
Previous results: large intersecting families are close to stars
Recent trends require study of nearly-intersecting families:
Families with relatively few disjoint pairs
Useful for studying supersaturation, probabilistic versions
Question (Removal)
What can we say about the structure of set families with few
disjoint pairs?
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Removal lemma
Theorem (Friedgut–Regev)
Let ζ > 0, and ζn ≤ k ≤ 12 − ζ n. For every ε > 0 there is a
n−k
δ > 0 such that if F ⊆ [n]
k , then F can
k has dp(F) ≤ δ |F|
be made intersecting by removing at most ε kn sets.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Removal lemma
Theorem (Friedgut–Regev)
Let ζ > 0, and ζn ≤ k ≤ 12 − ζ n. For every ε > 0 there is a
n−k
δ > 0 such that if F ⊆ [n]
k , then F can
k has dp(F) ≤ δ |F|
be made intersecting by removing at most ε kn sets.
Remarks
“Few disjoint pairs ⇒ ε-close to intersecting”
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Removal lemma
Theorem (Friedgut–Regev)
Let ζ > 0, and ζn ≤ k ≤ 12 − ζ n. For every ε > 0 there is a
n−k
δ > 0 such that if F ⊆ [n]
k , then F can
k has dp(F) ≤ δ |F|
be made intersecting by removing at most ε kn sets.
Remarks
“Few disjoint pairs ⇒ ε-close to intersecting”
Works for any F, regardless of closest intersecting family
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Our removal lemma
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that if 2 ≤ k < 21 n and
[n]
max{|α| , β} ≤ c · n−2k
n , then for every F ⊂ k with
n−1
n−1 n−k −1
|F| = (1 − α)
and dp(F) ≤ β
,
k −1
k −1
k −1
n−1
n
there is some star S such that |F∆S| ≤ C (α + 2β) n−2k
k−1 .
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Our removal lemma
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that if 2 ≤ k < 21 n and
[n]
max{|α| , β} ≤ c · n−2k
n , then for every F ⊂ k with
n−1
n−1 n−k −1
|F| = (1 − α)
and dp(F) ≤ β
,
k −1
k −1
k −1
n−1
n
there is some star S such that |F∆S| ≤ C (α + 2β) n−2k
k−1 .
Remarks
“Large nearly-intersecting families are close to stars”
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Our removal lemma
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that if 2 ≤ k < 21 n and
[n]
max{|α| , β} ≤ c · n−2k
n , then for every F ⊂ k with
n−1
n−1 n−k −1
|F| = (1 − α)
and dp(F) ≤ β
,
k −1
k −1
k −1
n−1
n
there is some star S such that |F∆S| ≤ C (α + 2β) n−2k
k−1 .
Remarks
“Large nearly-intersecting families are close to stars”
Works for all k, but only for large families
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Our removal lemma
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that if 2 ≤ k < 21 n and
[n]
max{|α| , β} ≤ c · n−2k
n , then for every F ⊂ k with
n−1
n−1 n−k −1
|F| = (1 − α)
and dp(F) ≤ β
,
k −1
k −1
k −1
n−1
n
there is some star S such that |F∆S| ≤ C (α + 2β) n−2k
k−1 .
Remarks
“Large nearly-intersecting families are close to stars”
Works for all k, but only for large families
Provides sharp quantitative bounds
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Observation
Independent sets ↔ intersecting families
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Observation
Independent sets ↔ intersecting families
Theorem (Erdős–Ko–Rado, 1961)
n−1
For k ≤ 12 n, α (KG (n, k)) = k−1
.
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Observation
KGp (n, k) ⊆ KG (n, k) ⇒ α(KGp (n, k)) ≥
n−1
k−1
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
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.
Observation
n−1
k−1
KGp (n, k) ⊆ KG (n, k) ⇒ α(KGp (n, k)) ≥
Question (Sparse Erdős–Ko–Rado)
For which p do we have α(KGp (n, k)) =
n−1
k−1
?
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Simulation: KG (5, 2)
12
35
45
34
13
25
14
23
0
24
15
1
p p p p p
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Natural threshold
Given star S, set F ∈
/ S:
n−k−1
k−1
edges from F to S
n−k−1
⇒ P(S ∪ {F } independent) = (1 − p)( k−1 )
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Natural threshold
Given star S, set F ∈
/ S:
n−k−1
k−1
edges from F to S
n−k−1
⇒ P(S ∪ {F } independent) = (1 − p)( k−1 )
n possible stars S,
⇒ threshold pc :=
n−1
sets
k
log(n(n−1
k ))
(n−k−1
k−1 )
F ∈
/S
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Natural threshold
Given star S, set F ∈
/ S:
n−k−1
k−1
edges from F to S
n−k−1
⇒ P(S ∪ {F } independent) = (1 − p)( k−1 )
n possible stars S,
⇒ threshold pc :=
n−1
sets
k
log(n(n−1
k ))
F ∈
/S
(n−k−1
k−1 )
p < (1 − ε)pc ⇒ w.h.p. stars not maximal independent sets
n−1
⇒ α(KGp (n, k)) > k−1
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Natural threshold
Given star S, set F ∈
/ S:
n−k−1
k−1
edges from F to S
n−k−1
⇒ P(S ∪ {F } independent) = (1 − p)( k−1 )
n possible stars S,
⇒ threshold pc :=
n−1
sets
k
log(n(n−1
k ))
F ∈
/S
(n−k−1
k−1 )
p < (1 − ε)pc ⇒ w.h.p. stars not maximal independent sets
n−1
⇒ α(KGp (n, k)) > k−1
p > (1 + ε)pc ⇒ w.h.p. stars are maximal independent sets
Can there be larger independent sets?
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Previous results
Theorem (Bollobás–Narayanan–Raigorodskii, 2014+)
Fix ε > 0, and let 2 ≤ k = o(n1/3 ). If p ≥ (1 + ε)pc ,
n−1
P α(KGp (n, k)) =
→1
k −1
as n → ∞. Moreover, with high probability the only maximum
independent sets are stars.
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Previous results
Theorem (Bollobás–Narayanan–Raigorodskii, 2014+)
Fix ε > 0, and let 2 ≤ k = o(n1/3 ). If p ≥ (1 + ε)pc ,
n−1
P α(KGp (n, k)) =
→1
k −1
as n → ∞. Moreover, with high probability the only maximum
independent sets are stars.
Theorem (Balogh–Bollobás–Narayanan, 2014+)
For every ζ > 0, there is a constant c = c(ζ) > 0 such that if
k ≤ 12 − ζ n, then, as n → ∞,
n−1
n − 1 −c
P α(KGp (n, k)) =
→ 1 if p ≥
.
k −1
k −1
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Our result
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that
n−1
P α(KGp (n, k)) =
→1
k −1
if either of the following hold:
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Our result
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that
n−1
P α(KGp (n, k)) =
→1
k −1
if either of the following hold:
(i) k ≤ cn and p ≥ (1 + ε)pc , for any ε = ω(k −1 ), or
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Our result
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that
n−1
P α(KGp (n, k)) =
→1
k −1
if either of the following hold:
(i) k ≤ cn and p ≥ (1 + ε)pc , for any ε = ω(k −1 ), or
(ii) k ≤ 12 (n − 3) and p ≥ C ·
n
n−2k
· pc .
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Our result
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that
n−1
P α(KGp (n, k)) =
→1
k −1
if either of the following hold:
(i) k ≤ cn and p ≥ (1 + ε)pc , for any ε = ω(k −1 ), or
(ii) k ≤ 12 (n − 3) and p ≥ C ·
n
n−2k
· pc .
Moreover, in both cases the stars are with high probability the only
maximum independent sets.
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Proof
Need to check
there are no non-star independent sets F of
n−1
size k−1
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Proof
Need to check
there are no non-star independent sets F of
n−1
size k−1
Simple union bound:
P (large non-star independent set) ≤
X
F
(1 − p)dp(F ) .
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Proof
Need to check
there are no non-star independent sets F of
n−1
size k−1
Simple union bound:
P (large non-star independent set) ≤
X
F
If dp(F) is large, contribution is insignificant
(1 − p)dp(F ) .
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Proof
Need to check
there are no non-star independent sets F of
n−1
size k−1
Simple union bound:
P (large non-star independent set) ≤
X
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
(1 − p)dp(F ) .
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Proof
Need to check
there are no non-star independent sets F of
n−1
size k−1
Simple union bound:
P (large non-star independent set) ≤
X
(1 − p)dp(F ) .
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
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Statement
Theorem (D.–Tran, 2015+)
There are constants c, C > 0 such that if 2 ≤ k < 21 n and
[n]
max{|α| , β} ≤ c · n−2k
n , then for every F ⊂ k with
n−1
n−1 n−k −1
|F| = (1 − α)
and dp(F) ≤ β
,
k −1
k −1
k −1
n−1
n
there is some star S such that |F∆S| ≤ C (α + 2β) n−2k
k−1 .
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Spectral framework
ConsiderPfunctions f : [n]
k → R as f (x1 , x2 , . . . , xn ) defined
on {x : i xi = k} or as vectors (f (F ))F ∈([n])
k
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Spectral framework
ConsiderPfunctions f : [n]
k → R as f (x1 , x2 , . . . , xn ) defined
on {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 T Af = 2dp(F).
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Spectral framework
ConsiderPfunctions f : [n]
k → R as f (x1 , x2 , . . . , xn ) defined
on {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 T Af = 2dp(F).
Definition (Affine functions)
Pn
A function f : [n]
i=1 ai xi for some
k → R is affine if f (x) = a0 +
coefficients a0 , a1 , . . . , an .
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Lemma 1
Lemma
Suppose 2 ≤ k < 12 n, and F ⊆ [n]
is a family with
k
n−1
n−1 n−k−1
|F| = (1 − α) k−1
and dp(F) ≤ β k−1
. If
k−1
[n]
f : k → {0, 1} is the characteristic function of F, there is some
k
2
affine function g : [n]
k → R with kf − g k ≤ (α + 2β) n−2k .
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Lemma 1
Lemma
Suppose 2 ≤ k < 12 n, and F ⊆ [n]
is a family with
k
n−1
n−1 n−k−1
|F| = (1 − α) k−1
and dp(F) ≤ β k−1
. If
k−1
[n]
f : k → {0, 1} is the characteristic function of F, there is some
k
2
affine function g : [n]
k → R with kf − g k ≤ (α + 2β) n−2k .
Remarks
Lovász (1979): determined spectrum of Kneser graphs
Eigenspaces corresponding to two most significant eigenvalues
are precisely the affine functions
Erdős–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 ≤ 21 n and
k
ε < 128n
, and f : [n]
k → {0, 1} is ε-close to an affine
n function,
o
√
then there is some S ⊂ [n] of size |S| ≤ t = max 1, Cnk
that either f or 1 − f is (C ε)-close to maxi∈S xi .
ε
such
Erdős–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 ≤ 21 n and
k
ε < 128n
, and f : [n]
k → {0, 1} is ε-close to an affine
n function,
o
√
then there is some S ⊂ [n] of size |S| ≤ t = max 1, Cnk
that either f or 1 − f is (C ε)-close to maxi∈S xi .
ε
such
Remarks
“Boolean + close to affine ⇒ almost determined by few
variables”
Version of the Friedgut–Kalai–Naor theorem for these uniform
slices of the Boolean cube
[n]\[s]
Gs = [n]
: characteristic function gs = maxi∈[s] xi .
k \
k
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Counting
Two lemmas ⇒ F is close to Gs or Gs for some 0 ≤ s ≤ t
Conclusion
Erdős–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
Erdős–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!
Erdős–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
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Open questions
Sharp threshold for sparse Erdős–Ko–Rado theorem when
k ∼ 12 n: is p ≥ (1 + ε)pc always sufficient?
Conclusion
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Open questions
Sharp threshold for sparse Erdős–Ko–Rado theorem when
k ∼ 12 n: is p ≥ (1 + ε)pc always sufficient?
Hitting time: if we randomly
remove edges from the Kneser
n−1
graph, does α(G ) > k−1 occur precisely when a star can be
extended?
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion
Open questions
Sharp threshold for sparse Erdős–Ko–Rado theorem when
k ∼ 12 n: is p ≥ (1 + ε)pc always sufficient?
Hitting time: if we randomly
remove edges from the Kneser
n−1
graph, does α(G ) > k−1 occur precisely when a star can be
extended?
Other applications of intersecting removal lemmas?
Erdős–Ko–Rado
Removal Lemmas
Sparse EKR
Removal Proof
Conclusion