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Coloring Simple Hypergraphs Dhruv Mubayi Department of Mathematics, Statistics and Computer Science University of Illinois Chicago May 13, 2011 Dhruv Mubayi Coloring Simple Hypergraphs A Puzzle Problem Let n 2 and suppose that S [ n

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Coloring Simple Hypergraphs

Dhruv Mubayi Department of Mathematics, Statistics and Computer Science University of Illinois Chicago May 13, 2011

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 2

A Puzzle

Problem Let n ≥ 2 and suppose that S ⊂ [n] × [n] with |S| ≥ 2n − 1. Then some three points in S determine a right angle.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 3

A Puzzle

Problem Let n ≥ 2 and suppose that S ⊂ [n] × [n] with |S| ≥ 2n − 1. Then some three points in S determine a right angle. If true, then sharp by letting S = {(x, y) : x = 1 or y = 1} \ (1, 1).

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 4

A Puzzle

Problem Let n ≥ 2 and suppose that S ⊂ [n] × [n] with |S| ≥ 2n − 1. Then some three points in S determine a right angle. If true, then sharp by letting S = {(x, y) : x = 1 or y = 1} \ (1, 1). What about 3-dimensions?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 5

Triangles in the Plane

Heilbronn Problem/Conjecture (1947) How large is the smallest triangle among n points in general position in the unit square?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 6

Triangles in the Plane

Heilbronn Problem/Conjecture (1947) How large is the smallest triangle among n points in general position in the unit square? S – collection of n points in general position in the unit square T(S) = area of smallest triangle T(n) = max

S

T(S)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 7

Trivial T(n) < c/n

Dhruv Mubayi Coloring Simple Hypergraphs

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Trivial T(n) < c/n Partition square into n/3 smaller squares of side length

  • 3/n

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 9

Trivial T(n) < c/n Partition square into n/3 smaller squares of side length

  • 3/n

Observation (Erd˝

  • s)

T(n) > c/n2

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 10

Trivial T(n) < c/n Partition square into n/3 smaller squares of side length

  • 3/n

Observation (Erd˝

  • s)

T(n) > c/n2 Explicit and probabilistic constructions exist

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 11

Trivial T(n) < c/n Partition square into n/3 smaller squares of side length

  • 3/n

Observation (Erd˝

  • s)

T(n) > c/n2 Explicit and probabilistic constructions exist Conjecture T(n) = Θ(1/n2)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 12

Upper Bounds

Dhruv Mubayi Coloring Simple Hypergraphs

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Upper Bounds Roth (1951) 1 n√log log n

Dhruv Mubayi Coloring Simple Hypergraphs

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Upper Bounds Roth (1951) 1 n√log log n Schmidt (1971) 1 n√log n

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 15

Upper Bounds Roth (1951) 1 n√log log n Schmidt (1971) 1 n√log n Roth (1972) 1 n1.117...+o(1) , 1.117 . . . = 17 − √ 65 8

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 16

Upper Bounds Roth (1951) 1 n√log log n Schmidt (1971) 1 n√log n Roth (1972) 1 n1.117...+o(1) , 1.117 . . . = 17 − √ 65 8 Koml´

  • s-Pintz-Szemer´

edi (1982) 1 n1.142...+o(1)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 17

Upper Bounds Roth (1951) 1 n√log log n Schmidt (1971) 1 n√log n Roth (1972) 1 n1.117...+o(1) , 1.117 . . . = 17 − √ 65 8 Koml´

  • s-Pintz-Szemer´

edi (1982) 1 n1.142...+o(1) Lower Bound Koml´

  • s-Pintz-Szemer´

edi (1982) T(n) > c log n n2

Dhruv Mubayi Coloring Simple Hypergraphs

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Number Theory - Infinite Sidon Sets

Definition S is a Sidon set if its pairwise sums are all distinct

Dhruv Mubayi Coloring Simple Hypergraphs

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Number Theory - Infinite Sidon Sets

Definition S is a Sidon set if its pairwise sums are all distinct Greedy Algorithm shows that there exists an infinite S with |S ∩ [n]| > cn1/3 for all n

Dhruv Mubayi Coloring Simple Hypergraphs

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Number Theory - Infinite Sidon Sets

Definition S is a Sidon set if its pairwise sums are all distinct Greedy Algorithm shows that there exists an infinite S with |S ∩ [n]| > cn1/3 for all n Ajtai-Koml´

  • s-Szemer´

edi (1981) (n log n)1/3

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 21

Number Theory - Infinite Sidon Sets

Definition S is a Sidon set if its pairwise sums are all distinct Greedy Algorithm shows that there exists an infinite S with |S ∩ [n]| > cn1/3 for all n Ajtai-Koml´

  • s-Szemer´

edi (1981) (n log n)1/3 Ruzsa (1998) n

√ 2−1−o(1)

Dhruv Mubayi Coloring Simple Hypergraphs

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Number Theory - Infinite Sidon Sets

Definition S is a Sidon set if its pairwise sums are all distinct Greedy Algorithm shows that there exists an infinite S with |S ∩ [n]| > cn1/3 for all n Ajtai-Koml´

  • s-Szemer´

edi (1981) (n log n)1/3 Ruzsa (1998) n

√ 2−1−o(1)

Conjecture (Erd˝

  • s)

n1/2−ǫ

Dhruv Mubayi Coloring Simple Hypergraphs

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Coding Theory

Fix k, r ≥ 2. Let A be an n × M matrix over Z2 with k one’s in each column every r columns linearly independent over Z2 (i.e. every set of at most r column vectors does not sum to 0)

Dhruv Mubayi Coloring Simple Hypergraphs

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Coding Theory

Fix k, r ≥ 2. Let A be an n × M matrix over Z2 with k one’s in each column every r columns linearly independent over Z2 (i.e. every set of at most r column vectors does not sum to 0) M := M(n, k, r) = maximum number of columns in A

Dhruv Mubayi Coloring Simple Hypergraphs

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Coding Theory

Fix k, r ≥ 2. Let A be an n × M matrix over Z2 with k one’s in each column every r columns linearly independent over Z2 (i.e. every set of at most r column vectors does not sum to 0) M := M(n, k, r) = maximum number of columns in A In other words, M is the maximum length of a binary linear code with minimum distance at least r + 1 and parity check matrix with n rows and every coordinate having at most k check equations.

Dhruv Mubayi Coloring Simple Hypergraphs

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Lefmann-Pudlak-Savick´ y (1997) M(n, k, r) > cn

kr 2(r−1) Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 27

Lefmann-Pudlak-Savick´ y (1997) M(n, k, r) > cn

kr 2(r−1)

Results of Frankl-F¨ uredi on union closed families yield M(n, k, 4) < cn

⌈4k/3⌉ 2

, so when k ≡ 0 (mod 3), M(n, k, 4) = Θ(n

2k 3 ) Dhruv Mubayi Coloring Simple Hypergraphs

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Lefmann-Pudlak-Savick´ y (1997) M(n, k, r) > cn

kr 2(r−1)

Results of Frankl-F¨ uredi on union closed families yield M(n, k, 4) < cn

⌈4k/3⌉ 2

, so when k ≡ 0 (mod 3), M(n, k, 4) = Θ(n

2k 3 )

Kretzberg-Hofmeister-Lefmann (1999) If r ≥ 4 is even, gcd(r − 1, k) = 1, then M(n, k, r) > cn

kr 2(r−1) (log n) 1 k−1 . Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 29

Lefmann-Pudlak-Savick´ y (1997) M(n, k, r) > cn

kr 2(r−1)

Results of Frankl-F¨ uredi on union closed families yield M(n, k, 4) < cn

⌈4k/3⌉ 2

, so when k ≡ 0 (mod 3), M(n, k, 4) = Θ(n

2k 3 )

Kretzberg-Hofmeister-Lefmann (1999) If r ≥ 4 is even, gcd(r − 1, k) = 1, then M(n, k, r) > cn

kr 2(r−1) (log n) 1 k−1 .

Naor-Verstra¨ ete (2009) Improvements for different ranges of k, r; connections to extremal graph theory

Dhruv Mubayi Coloring Simple Hypergraphs

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Independent Sets in Hypergraphs

Let k ≥ 2 be fixed, n → ∞

Dhruv Mubayi Coloring Simple Hypergraphs

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Independent Sets in Hypergraphs

Let k ≥ 2 be fixed, n → ∞ k-uniform hypergraph – edges have size k

Dhruv Mubayi Coloring Simple Hypergraphs

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Independent Sets in Hypergraphs

Let k ≥ 2 be fixed, n → ∞ k-uniform hypergraph – edges have size k independent set – vertex subset that contains no edge

Dhruv Mubayi Coloring Simple Hypergraphs

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Independent Sets in Hypergraphs

Let k ≥ 2 be fixed, n → ∞ k-uniform hypergraph – edges have size k independent set – vertex subset that contains no edge α(H) – maximum size of an independent set

Dhruv Mubayi Coloring Simple Hypergraphs

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Independent Sets in Hypergraphs

Let k ≥ 2 be fixed, n → ∞ k-uniform hypergraph – edges have size k independent set – vertex subset that contains no edge α(H) – maximum size of an independent set Example If H = K k

n =

[n]

k

  • , then α(H) = k − 1.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 35

Theorem Let H be a k-uniform hypergraph with average degree d. Then α(H) > ck n d1/(k−1) .

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem Let H be a k-uniform hypergraph with average degree d. Then α(H) > ck n d1/(k−1) .

  • Proof. Pick vertices randomly with appropriate probability p;

delete a vertex in each edge among the chosen vertices.

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem Let H be a k-uniform hypergraph with average degree d. Then α(H) > ck n d1/(k−1) .

  • Proof. Pick vertices randomly with appropriate probability p;

delete a vertex in each edge among the chosen vertices. Example Let H = K k

n , then d =

n−1

k−1

  • = Θ(nk−1), d

1 k−1 = Θ(n) and

α(H) = k − 1 = Θ(1)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 38

2−cycle 3−cycle

✁ ✂✄ ☎✆ ✝✞ ✟✠ ✡☛ ☞✌ ✍✎

4−cycle

girth g – no cycle of length less than g simple – girth 3 or no 2-cycle

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1).

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1). Theorem (Duke-Lefmann-R¨

  • dl 1995, Conjecture (Spencer 1990))

Same conclusion holds as long as H is simple.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 41

Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1).

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1). T(n) > c log n n2

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1). T(n) > c log n n2 |S| > c(n log n)1/3 (Improved by Ruzsa)

Dhruv Mubayi Coloring Simple Hypergraphs

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Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1). T(n) > c log n n2 |S| > c(n log n)1/3 (Improved by Ruzsa) M(n, k, r) > cn

kr 2(r−1) (log n) 1 k−1 . Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 45

Theorem (Koml´

  • s-Pintz-Szemer´

edi k = 3, Ajtai-Koml´

  • s-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 α(H) > c n ∆1/(k−1) (log ∆)1/(k−1). T(n) > c log n n2 |S| > c(n log n)1/3 (Improved by Ruzsa) M(n, k, r) > cn

kr 2(r−1) (log n) 1 k−1 .

Many other applications in combinatorics

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 46

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 47

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1 What if G is triangle-free?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 48

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1 What if G is triangle-free? Borodin-Kostochka: χ(G) ≤ 2 3(∆ + 2)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 49

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1 What if G is triangle-free? Borodin-Kostochka: χ(G) ≤ 2 3(∆ + 2) Since χ(G) ≥ n/α(G), what about independence number?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 50

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1 What if G is triangle-free? Borodin-Kostochka: χ(G) ≤ 2 3(∆ + 2) Since χ(G) ≥ n/α(G), what about independence number? Ajtai-Koml´

  • s-Szemer´

edi, Shearer: α(G) > c n ∆ log ∆

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 51

Graph Coloring

∆ = ∆(G) = max degree of G Greedy Algorithm: χ(G) ≤ ∆ + 1 Brook’s Theorem: χ(G) ≤ ∆ unless G = K∆+1 or G = C2r+1 What if G is triangle-free? Borodin-Kostochka: χ(G) ≤ 2 3(∆ + 2) Since χ(G) ≥ n/α(G), what about independence number? Ajtai-Koml´

  • s-Szemer´

edi, Shearer: α(G) > c n ∆ log ∆ Easy consequence: R(3, t) < c t2 log t

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 52

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

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SLIDE 53

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

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SLIDE 54

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

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SLIDE 55

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

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SLIDE 56

New Result

Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = ck such that every k-uniform simple H with maximum degree ∆ has χ(H) < c

  • ∆

log ∆

  • 1

k−1

.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 57

New Result

Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = ck such that every k-uniform simple H with maximum degree ∆ has χ(H) < c

  • ∆

log ∆

  • 1

k−1

. Bound without log ∆ factor is easy from the Local Lemma

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 58

New Result

Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = ck such that every k-uniform simple H with maximum degree ∆ has χ(H) < c

  • ∆

log ∆

  • 1

k−1

. 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

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SLIDE 59

New Result

Theorem (Frieze-M) Let k ≥ 3 be fixed. Then there exists c = ck such that every k-uniform simple H with maximum degree ∆ has χ(H) < c

  • ∆

log ∆

  • 1

k−1

. 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

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SLIDE 60

Semi-Random or “Nibble” Method

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 61

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

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SLIDE 62

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs

Dhruv Mubayi Coloring Simple Hypergraphs

slide-63
SLIDE 63

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (1985) result on hypergraph matchings

Dhruv Mubayi Coloring Simple Hypergraphs

slide-64
SLIDE 64

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (1985) result on hypergraph matchings

Pippenger-Spencer (1989) result of hypergraph edge-coloring

Dhruv Mubayi Coloring Simple Hypergraphs

slide-65
SLIDE 65

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (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

slide-66
SLIDE 66

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (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

slide-67
SLIDE 67

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (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

slide-68
SLIDE 68

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¨

  • dl’s proof (1985) of the Erd˝
  • s-Hanani conjecture on

asymptotically good designs Frankl-R¨

  • dl (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

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SLIDE 69

More Tools

Concentration Inequalities Hoeffding/Chernoff Talagrand Local Lemma Kim-Vu polynomial concentration takes care of dependencies

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 70

More Tools

Concentration Inequalities Hoeffding/Chernoff Talagrand Local Lemma Kim-Vu polynomial concentration takes care of dependencies Suppose X1, . . . , Xt are binomially distributed random variables with E( Xi) = µ, and they are “almost” independent. Then P

  • i

Xi − µ

  • > ǫµ
  • < e−cǫµ.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 71

Setup

x u v u 1 u 2 H U W

colored uncolored

H

t t t

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 72

The Algorithm (k = 3)

C = [q] – set of colors

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 73

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 74

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices Ht = H[Ut] – subgraph of H induced by Ut

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 75

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices Ht = H[Ut] – subgraph of H induced by Ut W t = V \ Ut – set of currently colored vertices

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 76

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices Ht = H[Ut] – subgraph of H induced by Ut W t = V \ Ut – set of currently colored vertices Ht

2 – colored graph

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 77

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices Ht = H[Ut] – subgraph of H induced by Ut W t = V \ Ut – set of currently colored vertices Ht

2 – colored graph

pt

u ∈ [0, 1]C, u ∈ Ut – vector of probabilities of colors

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 78

The Algorithm (k = 3)

C = [q] – set of colors Ut – set of currently uncolored vertices Ht = H[Ut] – subgraph of H induced by Ut W t = V \ Ut – set of currently colored vertices Ht

2 – colored graph

pt

u ∈ [0, 1]C, u ∈ Ut – vector of probabilities of colors

p0

u = (1/q, . . . , 1/q) – initial color vector

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 79

x u v u 1 u 2 H U W

colored uncolored

H

t t t

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 80

For u ∈ U, c ∈ [q], tentatively activate c at u with probability Θ · pu(c).

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 81

For u ∈ U, c ∈ [q], tentatively activate c at u with probability Θ · pu(c). A color is lost at u if either there is an edge uu1u2 such that c is tentatively activated at u1 and u2, or

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 82

For u ∈ U, c ∈ [q], tentatively activate c at u with probability Θ · pu(c). A color is lost at u if either there is an edge uu1u2 such that c is tentatively activated at u1 and u2, or x has been colored with c and c has been tentatively activated at v

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 83

For u ∈ U, c ∈ [q], tentatively activate c at u with probability Θ · pu(c). A color is lost at u if either there is an edge uu1u2 such that c is tentatively activated at u1 and u2, or x has been colored with c and c has been tentatively activated at v In this case pu(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

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SLIDE 84

For u ∈ U, c ∈ [q], tentatively activate c at u with probability Θ · pu(c). A color is lost at u if either there is an edge uu1u2 such that c is tentatively activated at u1 and u2, or x has been colored with c and c has been tentatively activated at v In this case pu(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 pu are updated in a (complicated) way to maintain certain properties of Ht = H[U]

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 85

Parameters (k = 3)

During the process, we must choose update values to maintain the values of certain parameters:

  • c pu(c) ∼ 1

euvw =

c pu(c)pv(c)pw(c) ≪ log ∆ ∆

deg(v) ≤

  • 1 −

1 log ∆

t ∆ ∼ e−t/ 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

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SLIDE 86

What next?

Independence number of locally sparse Graphs Let G contain no K4

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 87

What next?

Independence number of locally sparse Graphs Let G contain no K4 Ajtai-Erd˝

  • s-Koml´
  • s-Szemer´

edi (1981) α(G) > c n ∆ log log ∆

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 88

What next?

Independence number of locally sparse Graphs Let G contain no K4 Ajtai-Erd˝

  • s-Koml´
  • s-Szemer´

edi (1981) α(G) > c n ∆ log log ∆ Shearer (1995) α(G) > c n ∆ log ∆ log log ∆

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 89

What next?

Independence number of locally sparse Graphs Let G contain no K4 Ajtai-Erd˝

  • s-Koml´
  • s-Szemer´

edi (1981) α(G) > c n ∆ log log ∆ Shearer (1995) α(G) > c n ∆ log ∆ log log ∆ Major Open Conjecture (Erd˝

  • s et. al.)

α(G) > c n ∆ log ∆

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 90

More Optimism

Conjecture (Frieze-M) Let F be a fixed k-uniform hypergraph. Then there exists c = cF such that every F-free k-uniform hypergraph H with maximum degree ∆ satisfies χ(H) < c

  • ∆

log ∆

  • 1

k−1

.

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 91

More Optimism

Conjecture (Frieze-M) Let F be a fixed k-uniform hypergraph. Then there exists c = cF such that every F-free k-uniform hypergraph H with maximum degree ∆ satisfies χ(H) < c

  • ∆

log ∆

  • 1

k−1

. Weaker Conjecture: χ(H) = o(∆

1 k−1 ) Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 92

More Optimism

Conjecture (Frieze-M) Let F be a fixed k-uniform hypergraph. Then there exists c = cF such that every F-free k-uniform hypergraph H with maximum degree ∆ satisfies χ(H) < c

  • ∆

log ∆

  • 1

k−1

. Weaker Conjecture: χ(H) = o(∆

1 k−1 )

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

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SLIDE 93

An Application to Theoretical Computer Science

For ∆ fixed and n → ∞, a simple randomized algorithm yields α(H) = Ω

  • n

∆1/(k−1)

  • Dhruv Mubayi

Coloring Simple Hypergraphs

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SLIDE 94

An Application to Theoretical Computer Science

For ∆ fixed and n → ∞, a simple randomized algorithm yields α(H) = Ω

  • n

∆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. α(H) = O

  • n

log ∆ ∆ 1/(k−1) H is 2-colorable

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 95

An Application to Theoretical Computer Science

For ∆ fixed and n → ∞, a simple randomized algorithm yields α(H) = Ω

  • n

∆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. α(H) = O

  • n

log ∆ ∆ 1/(k−1) 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

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SLIDE 96

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 2 points which do not determine a right angle?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 97

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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

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SLIDE 98

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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

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SLIDE 99

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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 Erd˝

  • s (1977)

Ω(n2/3)

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 100

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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 Erd˝

  • s (1977)

Ω(n2/3) Elekes (2009) Ω

  • n

√log n

  • Dhruv Mubayi

Coloring Simple Hypergraphs

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SLIDE 101

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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 Erd˝

  • s (1977)

Ω(n2/3) Elekes (2009) Ω

  • n

√log n

  • Gy´

arf´ as-M Ω(n) assuming Frieze-M Conjecture holds for k = 3 and F = K 3

9

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 102

An Application to Discrete Geometry

Problem (Erd˝

  • s 1977)

Do n2 points in the plane always contain 2n − 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 Erd˝

  • s (1977)

Ω(n2/3) Elekes (2009) Ω

  • n

√log n

  • Gy´

arf´ as-M Ω(n) assuming Frieze-M Conjecture holds for k = 3 and F = K 3

9

What about in 3-dimensions?

Dhruv Mubayi Coloring Simple Hypergraphs

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SLIDE 103

Thank You

Dhruv Mubayi Coloring Simple Hypergraphs