Removal Lemma for Nearly-Intersecting Families Shagnik Das Freie Universit¨ at, Berlin, Germany August 25, 2015 Joint work with Tuan Tran
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Intersecting families Definition (Intersecting families) A family of sets F is intersecting if F 1 ∩ F 2 � = ∅ for all F 1 , F 2 ∈ F .
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Intersecting families Definition (Intersecting families) A family of sets F is intersecting if F 1 ∩ F 2 � = ∅ for all F 1 , F 2 ∈ F . Notation [ n ] = { 1 , 2 , . . . , n } - ground set for our set families � [ n ] � = { F ⊂ [ n ] : | F | = k } k
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Intersecting families Definition (Intersecting families) A family of sets F is intersecting if F 1 ∩ F 2 � = ∅ for all F 1 , F 2 ∈ F . Notation [ n ] = { 1 , 2 , . . . , n } - ground set for our set families � [ n ] � = { F ⊂ [ n ] : | F | = k } k dp ( F ) = |{ F , G ∈ F : F ∩ G = ∅}| F intersecting ⇔ dp ( F ) = 0
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Erd˝ os–Ko–Rado theorem Theorem (Erd˝ os–Ko–Rado, 1961) If k ≤ 1 � [ n ] � n − 1 � � 2 n, and F ⊆ is intersecting, then |F| ≤ . k − 1 k
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Erd˝ os–Ko–Rado theorem Theorem (Erd˝ os–Ko–Rado, 1961) If k ≤ 1 � [ n ] � n − 1 � � 2 n, and F ⊆ is intersecting, then |F| ≤ . k − 1 k If k > 1 � [ n ] � 2 n , itself is intersecting k If k < 1 2 n , unique extremal families are stars : all sets containing some fixed element i ∈ [ n ]
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Erd˝ os–Ko–Rado theorem Theorem (Erd˝ os–Ko–Rado, 1961) If k ≤ 1 � [ n ] � n − 1 � � 2 n, and F ⊆ is intersecting, then |F| ≤ . k − 1 k If k > 1 � [ n ] � 2 n , itself is intersecting k If k < 1 2 n , unique extremal families are stars : all sets containing some fixed element i ∈ [ n ] 1 A star with centre 1
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Stability Question (Stability) What can we say about the structure of large intersecting families?
Erd˝ os–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) � [ n ] If k < 1 2 n, and F ⊆ � is intersecting with k � n − 1 � n − k − 1 � � |F| > − + 1 , then F is contained in a star. k − 1 k − 1
Erd˝ os–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) � [ n ] If k < 1 2 n, and F ⊆ � is intersecting with k � n − 1 � n − k − 1 � � |F| > − + 1 , then F is contained in a star. k − 1 k − 1 Bound is best-possible, but . . . . . . the Hilton–Milner families have all but one set in a star.
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Robust stability Theorem (Friedgut, 2008) � 1 � Let ζ > 0 , and let ζ n ≤ k ≤ 2 − ζ n. There is some c = c ( ζ ) � [ n ] � n − 1 such that for every intersecting F ⊆ � with |F| ≥ (1 − ε ) � k k − 1 � n − 1 � there is a star S with |F \ S| ≤ c ε . k − 1
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Robust stability Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) � 1 � Let ζ > 0 , and let 2 ≤ k ≤ 2 − ζ n. There is some c = c ( ζ ) � [ n ] � n − 1 such that for every intersecting F ⊆ � with |F| ≥ (1 − ε ) � k k − 1 � n − 1 � there is a star S with |F \ S| ≤ c ε . k − 1 Much stronger result obtained when k = o ( n )
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Robust stability Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) � 1 � Let ζ > 0 , and let 2 ≤ k ≤ 2 − ζ n. There is some c = c ( ζ ) � [ n ] � n − 1 such that for every intersecting F ⊆ � with |F| ≥ (1 − ε ) � k k − 1 � n − 1 � there is a star S with |F \ S| ≤ c ε . 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 � [ n ] and ε n ≤ k ≤ 1 � 2 n − 1 , if F ⊆ is intersecting with k � � n − 1 1 − δ · n − 2 k � � |F| ≥ , then there is some star S with n k − 1 � n − 1 |F \ S| ≤ ε � . k − 1
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Robust stability Theorem (Friedgut, 2008; Dinur–Friedgut, 2009) � 1 � Let ζ > 0 , and let 2 ≤ k ≤ 2 − ζ n. There is some c = c ( ζ ) � [ n ] � n − 1 such that for every intersecting F ⊆ � with |F| ≥ (1 − ε ) � k k − 1 � n − 1 � there is a star S with |F \ S| ≤ c ε . 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 � [ n ] and ε n ≤ k ≤ 1 � 2 n − 1 , if F ⊆ is intersecting with k � � n − 1 1 − δ · n − 2 k � � |F| ≥ , then there is some star S with n k − 1 � n − 1 |F \ S| ≤ ε � . k − 1
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Nearly-intersecting families Previous results: large intersecting families are close to stars
Erd˝ os–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˝ os–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˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Removal lemma Theorem (Friedgut–Regev) � 1 � [ n ] Let ζ > 0 , and ζ n ≤ k ≤ 2 − ζ � n. If F ⊆ � is such that k dp ( F ) is small, then F can be made intersecting by removing few sets.
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Removal lemma Theorem (Friedgut–Regev) � 1 � [ n ] Let ζ > 0 , and ζ n ≤ k ≤ 2 − ζ � n. If F ⊆ � is such that k dp ( F ) is small, then F can be made intersecting by removing few sets. “Few disjoint pairs ⇒ ε -close to intersecting”
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Removal lemma Theorem (Friedgut–Regev) � 1 � Let ζ > 0 , and ζ n ≤ k ≤ 2 − ζ n. For every ε > 0 there is a � [ n ] � n − k � � δ > 0 such that if F ⊆ has dp ( F ) ≤ δ |F| , then F can k k � n � be made intersecting by removing at most ε sets. k “Few disjoint pairs ⇒ ε -close to intersecting”
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Removal lemma Theorem (Friedgut–Regev) � 1 � Let ζ > 0 , and ζ n ≤ k ≤ 2 − ζ n. For every ε > 0 there is a � [ n ] � n − k � � δ > 0 such that if F ⊆ has dp ( F ) ≤ δ |F| , then F can k k � n � be made intersecting by removing at most ε sets. k “Few disjoint pairs ⇒ ε -close to intersecting” Works for any F , regardless of closest intersecting family
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Our removal lemma Theorem (D.–Tran) � [ n ] If 2 ≤ k < 1 � 2 n, then for every F ⊂ with k � n − 1 � |F| close to and dp ( F ) small , k − 1 there is some star S such that |F ∆ S| is small.
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Our removal lemma Theorem (D.–Tran) � [ n ] If 2 ≤ k < 1 � 2 n, then for every F ⊂ with k � n − 1 � |F| close to and dp ( F ) small , k − 1 there is some star S such that |F ∆ S| is small. “Large nearly-intersecting families are close to stars”
Erd˝ os–Ko–Rado Removal Lemmas Sparse EKR Removal Proof Conclusion Our removal lemma Theorem (D.–Tran) � [ n ] If 2 ≤ k < 1 � 2 n, then for every F ⊂ with k � n − 1 � |F| close to and dp ( F ) small , k − 1 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
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