Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c Theorem (Duke-Lefmann-R¨ odl 1995, Conjecture (Spencer 1990)) Same conclusion holds as long as H is simple. Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c T ( n ) > c log n n 2 Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c T ( n ) > c log n n 2 | S | > c ( n log n ) 1 / 3 (Improved by Ruzsa) Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c T ( n ) > c log n n 2 | S | > c ( n log n ) 1 / 3 (Improved by Ruzsa) kr 1 2( r − 1) (log n ) k − 1 . M ( n , k , r ) > cn Dhruv Mubayi Coloring Simple Hypergraphs
Theorem (Koml´ os-Pintz-Szemer´ edi k = 3, Ajtai-Koml´ os-Pintz-Spencer-Szemer´ edi k ≥ 3 1982) Let k ≥ 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree ∆ . Then n ∆ 1 / ( k − 1) (log ∆) 1 / ( k − 1) . α ( H ) > c T ( n ) > c log n n 2 | S | > c ( n log n ) 1 / 3 (Improved by Ruzsa) kr 1 2( r − 1) (log n ) k − 1 . M ( n , k , r ) > cn Many other applications in combinatorics Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 What if G is triangle-free? Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 What if G is triangle-free? χ ( G ) ≤ 2 Borodin-Kostochka: 3(∆ + 2) Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 What if G is triangle-free? χ ( G ) ≤ 2 Borodin-Kostochka: 3(∆ + 2) Since χ ( G ) ≥ n /α ( G ), what about independence number? Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 What if G is triangle-free? χ ( G ) ≤ 2 Borodin-Kostochka: 3(∆ + 2) Since χ ( G ) ≥ n /α ( G ), what about independence number? α ( G ) > c n Ajtai-Koml´ os-Szemer´ edi, Shearer: ∆ log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
Graph Coloring ∆ = ∆( G ) = max degree of G Greedy Algorithm: χ ( G ) ≤ ∆ + 1 Brook’s Theorem: χ ( G ) ≤ ∆ unless G = K ∆+1 or G = C 2 r +1 What if G is triangle-free? χ ( G ) ≤ 2 Borodin-Kostochka: 3(∆ + 2) Since χ ( G ) ≥ n /α ( G ), what about independence number? α ( G ) > c n Ajtai-Koml´ os-Szemer´ edi, Shearer: ∆ log ∆ R (3 , t ) < c t 2 Easy consequence: log t Dhruv Mubayi Coloring Simple Hypergraphs
Question (Vizing 1968) What is the best possible bound on the chromatic number of a triangle-free graph G in terms of its maximum degree? Dhruv Mubayi Coloring Simple Hypergraphs
Question (Vizing 1968) What is the best possible bound on the chromatic number of a triangle-free graph G in terms of its maximum degree? Random graphs show that there exist triangle-free graphs G with ∆ χ ( G ) > c log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
Question (Vizing 1968) What is the best possible bound on the chromatic number of a triangle-free graph G in terms of its maximum degree? Random graphs show that there exist triangle-free graphs G with ∆ χ ( G ) > c log ∆ ∆ Kim (1995): If girth( G ) ≥ 5, then χ ( G ) < c log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
Question (Vizing 1968) What is the best possible bound on the chromatic number of a triangle-free graph G in terms of its maximum degree? Random graphs show that there exist triangle-free graphs G with ∆ χ ( G ) > c log ∆ ∆ Kim (1995): If girth( G ) ≥ 5, then χ ( G ) < c log ∆ Johansson (1997): If G is triangle-free, then ∆ χ ( G ) < c log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
New Result Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = c k such that every k-uniform simple H with maximum degree ∆ has 1 � ∆ � k − 1 χ ( H ) < c . log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
New Result Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = c k such that every k-uniform simple H with maximum degree ∆ has 1 � ∆ � k − 1 χ ( H ) < c . log ∆ Bound without log ∆ factor is easy from the Local Lemma Dhruv Mubayi Coloring Simple Hypergraphs
New Result Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = c k such that every k-uniform simple H with maximum degree ∆ has 1 � ∆ � k − 1 χ ( H ) < c . log ∆ Bound without log ∆ factor is easy from the Local Lemma Proof is independent of K-P-Sz and A-K-P-S-Sz (and D-L-R) so it gives a new proof of those results Dhruv Mubayi Coloring Simple Hypergraphs
New Result Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = c k such that every k-uniform simple H with maximum degree ∆ has 1 � ∆ � k − 1 χ ( H ) < c . log ∆ Bound without log ∆ factor is easy from the Local Lemma Proof is independent of K-P-Sz and A-K-P-S-Sz (and D-L-R) so it gives a new proof of those results The result is sharp apart from the constant c Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring Kahn (1990s) proved many results, list coloring using different approach to P-S Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring Kahn (1990s) proved many results, list coloring using different approach to P-S Kim (1995) graphs of girth five Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring Kahn (1990s) proved many results, list coloring using different approach to P-S Kim (1995) graphs of girth five Johansson (1997) additional new ideas for triangle-free graphs Dhruv Mubayi Coloring Simple Hypergraphs
Semi-Random or “Nibble” Method A-K-Sz, K-P-Sz and A-K-P-S-Sz (1980-82) were perhaps the first papers using this approach R¨ odl’s proof (1985) of the Erd˝ os-Hanani conjecture on asymptotically good designs Frankl-R¨ odl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring Kahn (1990s) proved many results, list coloring using different approach to P-S Kim (1995) graphs of girth five Johansson (1997) additional new ideas for triangle-free graphs Vu (2000+) extended Johansson’s ideas to more general situations Dhruv Mubayi Coloring Simple Hypergraphs
More Tools Concentration Inequalities Hoeffding/Chernoff Talagrand Local Lemma Kim-Vu polynomial concentration takes care of dependencies Dhruv Mubayi Coloring Simple Hypergraphs
More Tools Concentration Inequalities Hoeffding/Chernoff Talagrand Local Lemma Kim-Vu polynomial concentration takes care of dependencies Suppose X 1 , . . . , X t are binomially distributed random variables with E ( � X i ) = µ , and they are “almost” independent. Then �� � � � � � < e − c ǫ µ . P X i − µ � > ǫµ � � � � � i Dhruv Mubayi Coloring Simple Hypergraphs
Setup H u u 2 u 1 t H x v colored uncolored t W t U Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices H t = H [ U t ] – subgraph of H induced by U t Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices H t = H [ U t ] – subgraph of H induced by U t W t = V \ U t – set of currently colored vertices Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices H t = H [ U t ] – subgraph of H induced by U t W t = V \ U t – set of currently colored vertices H t 2 – colored graph Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices H t = H [ U t ] – subgraph of H induced by U t W t = V \ U t – set of currently colored vertices H t 2 – colored graph u ∈ [0 , 1] C , u ∈ U t – vector of probabilities of colors p t Dhruv Mubayi Coloring Simple Hypergraphs
The Algorithm ( k = 3) C = [ q ] – set of colors U t – set of currently uncolored vertices H t = H [ U t ] – subgraph of H induced by U t W t = V \ U t – set of currently colored vertices H t 2 – colored graph u ∈ [0 , 1] C , u ∈ U t – vector of probabilities of colors p t p 0 u = (1 / q , . . . , 1 / q ) – initial color vector Dhruv Mubayi Coloring Simple Hypergraphs
H u u 2 u 1 t H x v colored uncolored t W t U Dhruv Mubayi Coloring Simple Hypergraphs
For u ∈ U , c ∈ [ q ], tentatively activate c at u with probability Θ · p u ( c ) . Dhruv Mubayi Coloring Simple Hypergraphs
For u ∈ U , c ∈ [ q ], tentatively activate c at u with probability Θ · p u ( c ) . A color is lost at u if either there is an edge uu 1 u 2 such that c is tentatively activated at u 1 and u 2 , or Dhruv Mubayi Coloring Simple Hypergraphs
For u ∈ U , c ∈ [ q ], tentatively activate c at u with probability Θ · p u ( c ) . A color is lost at u if either there is an edge uu 1 u 2 such that c is tentatively activated at u 1 and u 2 , or x has been colored with c and c has been tentatively activated at v Dhruv Mubayi Coloring Simple Hypergraphs
For u ∈ U , c ∈ [ q ], tentatively activate c at u with probability Θ · p u ( c ) . A color is lost at u if either there is an edge uu 1 u 2 such that c is tentatively activated at u 1 and u 2 , or x has been colored with c and c has been tentatively activated at v In this case p u ( c ) = 0 for all further iterations Assign a permanent color to u if some color c is tentatively activated at u and is not lost Dhruv Mubayi Coloring Simple Hypergraphs
For u ∈ U , c ∈ [ q ], tentatively activate c at u with probability Θ · p u ( c ) . A color is lost at u if either there is an edge uu 1 u 2 such that c is tentatively activated at u 1 and u 2 , or x has been colored with c and c has been tentatively activated at v In this case p u ( c ) = 0 for all further iterations Assign a permanent color to u if some color c is tentatively activated at u and is not lost Parameters p u are updated in a (complicated) way to maintain certain properties of H t = H [ U ] Dhruv Mubayi Coloring Simple Hypergraphs
Parameters ( k = 3) During the process, we must choose update values to maintain the values of certain parameters: � c p u ( c ) ∼ 1 c p u ( c ) p v ( c ) p w ( c ) ≪ log ∆ e uvw = � ∆ � t � 1 ∆ ∼ e − t / log ∆ ∆ deg( v ) ≤ 1 − log ∆ Also, entropy is controlled; key new idea of Johansson; don’t need martingales, Hoeffding suffices Continue till t = log ∆ log log ∆ and then apply Local Lemma. Dhruv Mubayi Coloring Simple Hypergraphs
What next? Independence number of locally sparse Graphs Let G contain no K 4 Dhruv Mubayi Coloring Simple Hypergraphs
What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erd˝ os-Koml´ os-Szemer´ edi (1981) α ( G ) > c n ∆ log log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erd˝ os-Koml´ os-Szemer´ edi (1981) α ( G ) > c n ∆ log log ∆ Shearer (1995) α ( G ) > c n log ∆ ∆ log log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erd˝ os-Koml´ os-Szemer´ edi (1981) α ( G ) > c n ∆ log log ∆ Shearer (1995) α ( G ) > c n log ∆ ∆ log log ∆ Major Open Conjecture (Erd˝ os et. al.) α ( G ) > c n ∆ log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
More Optimism Conjecture (Frieze-M) Let F be a fixed k -uniform hypergraph. Then there exists c = c F such that every F -free k -uniform hypergraph H with maximum degree ∆ satisfies 1 � ∆ � k − 1 χ ( H ) < c . log ∆ Dhruv Mubayi Coloring Simple Hypergraphs
More Optimism Conjecture (Frieze-M) Let F be a fixed k -uniform hypergraph. Then there exists c = c F such that every F -free k -uniform hypergraph H with maximum degree ∆ satisfies 1 � ∆ � k − 1 χ ( H ) < c . log ∆ 1 k − 1 ) Weaker Conjecture: χ ( H ) = o (∆ Dhruv Mubayi Coloring Simple Hypergraphs
More Optimism Conjecture (Frieze-M) Let F be a fixed k -uniform hypergraph. Then there exists c = c F such that every F -free k -uniform hypergraph H with maximum degree ∆ satisfies 1 � ∆ � k − 1 χ ( H ) < c . log ∆ 1 k − 1 ) Weaker Conjecture: χ ( H ) = o (∆ Algorithms?? Convert our proof to a deterministic polynomial time algorithm that yields a coloring with c (∆ / log ∆) 1 / ( k − 1) colors Moser-Tardos results yield a randomized algorithm Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Theoretical Computer Science For ∆ fixed and n → ∞ , a simple randomized algorithm yields n � � α ( H ) = Ω ∆ 1 / ( k − 1) Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Theoretical Computer Science For ∆ fixed and n → ∞ , a simple randomized algorithm yields n � � α ( H ) = Ω ∆ 1 / ( k − 1) Theorem (Guruswami and Sinop 2010) It is NP-hard to distinguish between the following for a k-uniform hypergraph H with k ≥ 4 . � � 1 / ( k − 1) � � log ∆ α ( H ) = O n ∆ H is 2-colorable Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Theoretical Computer Science For ∆ fixed and n → ∞ , a simple randomized algorithm yields n � � α ( H ) = Ω ∆ 1 / ( k − 1) Theorem (Guruswami and Sinop 2010) It is NP-hard to distinguish between the following for a k-uniform hypergraph H with k ≥ 4 . � � 1 / ( k − 1) � � log ∆ α ( H ) = O n ∆ H is 2-colorable Moreover, the (log ∆) 1 / ( k − 1) factor above cannot be improved assuming P � = NP and the Frieze-M conjecture Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Discrete Geometry Problem (Erd˝ os 1977) Do n 2 points in the plane always contain 2 n − 2 points which do not determine a right angle? Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Discrete Geometry Problem (Erd˝ os 1977) Do n 2 points in the plane always contain 2 n − 2 points which do not determine a right angle? If true, then sharp (take [ n ] × [ n ] and use earlier Problem) Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Discrete Geometry Problem (Erd˝ os 1977) Do n 2 points in the plane always contain 2 n − 2 points which do not determine a right angle? If true, then sharp (take [ n ] × [ n ] and use earlier Problem) Lower bounds on the number of points Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Discrete Geometry Problem (Erd˝ os 1977) Do n 2 points in the plane always contain 2 n − 2 points which do not determine a right angle? If true, then sharp (take [ n ] × [ n ] and use earlier Problem) Lower bounds on the number of points Ω( n 2 / 3 ) Erd˝ os (1977) Dhruv Mubayi Coloring Simple Hypergraphs
An Application to Discrete Geometry Problem (Erd˝ os 1977) Do n 2 points in the plane always contain 2 n − 2 points which do not determine a right angle? If true, then sharp (take [ n ] × [ n ] and use earlier Problem) Lower bounds on the number of points Ω( n 2 / 3 ) Erd˝ os (1977) � n � Elekes (2009) Ω √ log n Dhruv Mubayi Coloring Simple Hypergraphs
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