Honors Combinatorics CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 6, Thursday, May 14, 2020 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs If an r -uniform hypergraph has ≤ 2 r − 10 Theorem (Erd˝ os) edges then it is 2-colorable. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs If an r -uniform hypergraph has ≤ 2 r − 10 Theorem (Erd˝ os) edges then it is 2-colorable. Proof. Let the edges be E 1 , . . . , E m . Color the vertices red/blue at random. Let B i be the event that E i becomes monochromatic ( bad event ). P ( B i ) = 2 2 r ∴ P ( coloring illegal ) = P ( � m i = 1 B i ) ≤ � m i = 1 P ( B i ) = 2 m 1 2 r ≤ 512 Not only does a good coloring exist, but 99.8% of the colorings works. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs QUESTION (Erd˝ os) What if we don’t limit the number of edges, only the degree of the vertices? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs QUESTION (Erd˝ os) What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs QUESTION (Erd˝ os) What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability? Close to 2 r ? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs QUESTION (Erd˝ os) What if we don’t limit the number of edges, only the degree of the vertices? What is the largest degree bound that will guarantee 2-colorability? Close to 2 r ? YES CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs Random coloring? Suppose degree ≤ 1 (edges are disjoint) — Probability of success? � m � 1 − 2 < e − 2 m / 2 r → 0 ∴ P ( coloring legal ) = 2 r as m → ∞ exponentially small. Random coloring will not work. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
2-colorable hypergraphs Random coloring? Suppose degree ≤ 1 (edges are disjoint) — Probability of success? � m � 1 − 2 < e − 2 m / 2 r → 0 ∴ P ( coloring legal ) = 2 r as m → ∞ exponentially small. Random coloring will not work. Lovász (1976): “Not so fast. Exponentially small but positive chance is still success.” CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ( V , F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event B v CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ( V , F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event B v Assumptions: 1. each B v independent of the set { B w | w not a neighbor of v } 2. ( ∀ v ∈ V )( deg ( v ) ≤ d ) 3. ( ∀ v ∈ V )( P ( B v ) ≤ 1 / ( 4 d )) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ( V , F ) graph (2-uniform hypergraph) each v ∈ V associated with bad event B v Assumptions: 1. each B v independent of the set { B w | w not a neighbor of v } 2. ( ∀ v ∈ V )( deg ( v ) ≤ d ) 3. ( ∀ v ∈ V )( P ( B v ) ≤ 1 / ( 4 d )) Conclusion: � P B v > 0 v ∈ V We have positive chance to avoid all the bad events. CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Independence from a set of events Let ǫ 1 , . . . , ǫ k ∈ { 1 , − 1 } . Events B 1 , . . . , B k define 2 k atoms B ǫ 1 1 ∩ · · · ∩ B ǫ k k where B 1 i = B i and B − 1 = B i . i CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Independence from a set of events Let ǫ 1 , . . . , ǫ k ∈ { 1 , − 1 } . Events B 1 , . . . , B k define 2 k atoms B ǫ 1 1 ∩ · · · ∩ B ǫ k k where B 1 i = B i and B − 1 = B i . i DEF: Event A and the set { B 1 , . . . , B k } of events are independent if A is independent of each atom of the B i . CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Independence from a set of events Let ǫ 1 , . . . , ǫ k ∈ { 1 , − 1 } . Events B 1 , . . . , B k define 2 k atoms B ǫ 1 1 ∩ · · · ∩ B ǫ k k where B 1 i = B i and B − 1 = B i . i DEF: Event A and the set { B 1 , . . . , B k } of events are independent if A is independent of each atom of the B i . DO: A and { B 1 , . . . , B k } are independent ⇐⇒ for every Boolean function f : { 0 , 1 } k → { 0 , 1 } , A is independent of f ( B 1 , . . . , B k ) . CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Conditional probabilities DEF: If P ( B ) > 0 then P ( A | B ) := P ( A ∩ B ) P ( B ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Conditional probabilities DEF: If P ( B ) > 0 then P ( A | B ) := P ( A ∩ B ) P ( B ) Lemma P ( A | C ∩ D ) = P ( A ∩ C | D ) P ( C | D ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Conditional probabilities DEF: If P ( B ) > 0 then P ( A | B ) := P ( A ∩ B ) P ( B ) Lemma P ( A | C ∩ D ) = P ( A ∩ C | D ) P ( C | D ) Proof. P ( A | C ∩ D ) = P ( A ∩ C ∩ D ) = P ( A ∩ C | D ) P ( D ) P ( C ∩ D ) P ( C | D ) P ( D ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ([ n ] , F ) graph; events B i ( i ∈ [ n ]) 1. each B i independent of the set { B j | j not a neighbor of i } 2. ( ∀ i ∈ [ n ])( deg ( i ) ≤ d ) � � P ( B i ) ≤ 1 3. ( ∀ i ∈ [ n ]) 4 d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ([ n ] , F ) graph; events B i ( i ∈ [ n ]) 1. each B i independent of the set { B j | j not a neighbor of i } 2. ( ∀ i ∈ [ n ])( deg ( i ) ≤ d ) � � P ( B i ) ≤ 1 3. ( ∀ i ∈ [ n ]) 4 d �� n � Then P i = 1 B i > 0 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ([ n ] , F ) graph; events B i ( i ∈ [ n ]) 1. each B i independent of the set { B j | j not a neighbor of i } 2. ( ∀ i ∈ [ n ])( deg ( i ) ≤ d ) � � P ( B i ) ≤ 1 3. ( ∀ i ∈ [ n ]) 4 d �� n � Then P i = 1 B i > 0 Proof by induction on n . n � � ≤ 1 � � Lemma P B 1 B i � 2 d � i = 2 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma G = ([ n ] , F ) graph; events B i ( i ∈ [ n ]) 1. each B i independent of the set { B j | j not a neighbor of i } 2. ( ∀ i ∈ [ n ])( deg ( i ) ≤ d ) � � P ( B i ) ≤ 1 3. ( ∀ i ∈ [ n ]) 4 d �� n � Then P i = 1 B i > 0 Proof by induction on n . n � � ≤ 1 � � Lemma P B 1 B i � 2 d � i = 2 Condition has positive probability by inductive hypothesis CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma Theorem � � P ( B i ) ≤ 1 �� n � ( ∀ i ∈ [ n ]) = ⇒ P i = 1 B i > 0 4 d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma Theorem � � P ( B i ) ≤ 1 �� n � ( ∀ i ∈ [ n ]) = ⇒ P i = 1 B i > 0 4 d Proof by induction on n . � n � ≤ 1 � � Lemma P B 1 B i � 2 d � i = 2 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma Theorem � � P ( B i ) ≤ 1 �� n � ( ∀ i ∈ [ n ]) = ⇒ P i = 1 B i > 0 4 d Proof by induction on n . � n � ≤ 1 � � Lemma P B 1 B i � 2 d � i = 2 Lemma = ⇒ Theorem CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma Theorem � � P ( B i ) ≤ 1 �� n � ( ∀ i ∈ [ n ]) = ⇒ P i = 1 B i > 0 4 d Proof by induction on n . � n � ≤ 1 � � Lemma P B 1 B i � 2 d � i = 2 Lemma = ⇒ Theorem Proof: F := B 2 ∩ · · · ∩ B n . � � 1 − 1 P ( B 1 ∩· · ·∩ B n ) = P ( B 1 ∩ F ) = P ( B 1 | F ) P ( F ) ≥ P ( F ) > 0 2 d QED CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma n � ≤ 1 � � � Lemma P B 1 B i � 2 d � i = 2 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
Lovász Local Lemma n � ≤ 1 � � � Lemma P B 1 B i � 2 d � i = 2 Proof. Induction on n . Neighbors of vertex 1: { 2 , . . . , k } C := B 2 ∩ · · · ∩ B k D := B k + 1 ∩ · · · ∩ B n n � = P ( B 1 | C ∩ D ) = P ( B 1 ∩ C | D ) ≤ 1 � � ? � P B 1 B i � P ( C | D ) 2 d � i = 2 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics
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