Coloring Simple Hypergraphs

Transcription

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

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.

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).

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?

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?

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 T (S) S

7 Trivial T (n) < c/n

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

9 Trivial T (n) < c/n Partition square into n/3 smaller squares of side length 3/n Observation (Erdős) T (n) > c/n 2

10 Trivial T (n) < c/n Partition square into n/3 smaller squares of side length 3/n Observation (Erdős) T (n) > c/n 2 Explicit and probabilistic constructions exist

11 Trivial T (n) < c/n Partition square into n/3 smaller squares of side length 3/n Observation (Erdős) T (n) > c/n 2 Explicit and probabilistic constructions exist Conjecture T (n) = Θ(1/n 2 )

12 Upper Bounds

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

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

15 Roth (1951) Schmidt (1971) Roth (1972) Upper Bounds 1 n log log n 1 n log n 1 n o(1), = 8

16 Roth (1951) Schmidt (1971) Roth (1972) Upper Bounds 1 n log log n 1 n log n 1 n o(1), Komlós-Pintz-Szemerédi (1982) = 8 1 n o(1)

17 Roth (1951) Schmidt (1971) Roth (1972) Upper Bounds 1 n log log n 1 n log n 1 n o(1), Komlós-Pintz-Szemerédi (1982) = 8 1 n o(1) Komlós-Pintz-Szemerédi (1982) Lower Bound T (n) > c log n n 2

18 Number Theory - Infinite Sidon Sets Definition S is a Sidon set if its pairwise sums are all distinct

19 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 for all n S [n] > cn 1/3

20 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 for all n S [n] > cn 1/3 Ajtai-Komlós-Szemerédi (1981) (n log n) 1/3

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 for all n S [n] > cn 1/3 Ajtai-Komlós-Szemerédi (1981) (n log n) 1/3 Ruzsa (1998) n 2 1 o(1)

22 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 for all n S [n] > cn 1/3 Ajtai-Komlós-Szemerédi (1981) (n log n) 1/3 Ruzsa (1998) n 2 1 o(1) Conjecture (Erdős) n 1/2 ɛ

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

24 Coding Theory Fix k, r 2. Let A be an n M matrix over Z 2 with k one s in each column every r columns linearly independent over Z 2 (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

25 Coding Theory Fix k, r 2. Let A be an n M matrix over Z 2 with k one s in each column every r columns linearly independent over Z 2 (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.

26 Lefmann-Pudlak-Savický (1997) M(n, k, r) > cn kr 2(r 1)

27 Lefmann-Pudlak-Savický (1997) M(n, k, r) > cn kr 2(r 1) Results of Frankl-Füredi 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 )

28 Lefmann-Pudlak-Savický (1997) M(n, k, r) > cn kr 2(r 1) Results of Frankl-Füredi 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 2(r 1) (log n) 1 kr k 1.

29 Lefmann-Pudlak-Savický (1997) M(n, k, r) > cn kr 2(r 1) Results of Frankl-Füredi 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 2(r 1) (log n) 1 kr k 1. Naor-Verstraëte (2009) Improvements for different ranges of k, r; connections to extremal graph theory

30 Independent Sets in Hypergraphs Let k 2 be fixed, n

31 Independent Sets in Hypergraphs Let k 2 be fixed, n k-uniform hypergraph edges have size k

32 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

33 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

34 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.

35 Theorem Let H be a k-uniform hypergraph with average degree d. Then n α(h) > c k d 1/(k 1).

36 Theorem Let H be a k-uniform hypergraph with average degree d. Then n α(h) > c k d 1/(k 1). Proof. Pick vertices randomly with appropriate probability p; delete a vertex in each edge among the chosen vertices.

37 Theorem Let H be a k-uniform hypergraph with average degree d. Then n α(h) > c k d 1/(k 1). Proof. Pick vertices randomly with appropriate probability p; delete a vertex in each edge among the chosen vertices. Example Let H = Kn k, then d = ( ) n 1 k 1 = Θ(n k 1 ), d 1 k 1 = Θ(n) and α(h) = k 1 = Θ(1)

38 2 cycle 3 cycle 4 cycle girth g no cycle of length less than g simple girth 3 or no 2-cycle

39 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1).

40 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1). Theorem (Duke-Lefmann-Rödl 1995, Conjecture (Spencer 1990)) Same conclusion holds as long as H is simple.

41 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1).

42 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1). T (n) > c log n n 2

43 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1). T (n) > c log n n 2 S > c(n log n) 1/3 (Improved by Ruzsa)

44 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1). T (n) > c log n n 2 S > c(n log n) 1/3 (Improved by Ruzsa) M(n, k, r) > cn 2(r 1) (log n) 1 kr k 1.

45 Theorem (Komlós-Pintz-Szemerédi k = 3, Ajtai-Komlós-Pintz-Spencer-Szemerédi k ) Let k 3 be fixed. Let H be a k-uniform hypergraph with girth at least 5 and (average) maximum degree. Then n α(h) > c 1/(k 1) (log )1/(k 1). T (n) > c log n n 2 S > c(n log n) 1/3 (Improved by Ruzsa) M(n, k, r) > cn 2(r 1) (log n) 1 kr k 1. Many other applications in combinatorics

46 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1

47 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1 What if G is triangle-free?

48 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1 What if G is triangle-free? Borodin-Kostochka: χ(g) 2 ( + 2) 3

49 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1 What if G is triangle-free? Borodin-Kostochka: χ(g) 2 ( + 2) 3 Since χ(g) n/α(g), what about independence number?

50 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1 What if G is triangle-free? Borodin-Kostochka: χ(g) 2 ( + 2) 3 Since χ(g) n/α(g), what about independence number? Ajtai-Komlós-Szemerédi, Shearer: α(g) > c n log

51 Graph Coloring = (G) = max degree of G Greedy Algorithm: χ(g) + 1 Brook s Theorem: χ(g) unless G = K +1 or G = C 2r+1 What if G is triangle-free? Borodin-Kostochka: χ(g) 2 ( + 2) 3 Since χ(g) n/α(g), what about independence number? Ajtai-Komlós-Szemerédi, Shearer: α(g) > c n log Easy consequence: R(3, t) < c t2 log t

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?

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

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

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

56 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 χ(h) < c ( ) 1 k 1. log

57 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 χ(h) < c ( ) 1 k 1. log Bound without log factor is easy from the Local Lemma

58 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 χ(h) < c ( ) 1 k 1. 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

59 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 χ(h) < c ( ) 1 k 1. 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

60 Semi-Random or Nibble Method

61 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach

62 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs

63 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (1985) result on hypergraph matchings

64 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (1985) result on hypergraph matchings Pippenger-Spencer (1989) result of hypergraph edge-coloring

65 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (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

66 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (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

67 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (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

68 Semi-Random or Nibble Method A-K-Sz, K-P-Sz and A-K-P-S-Sz ( ) were perhaps the first papers using this approach Rödl s proof (1985) of the Erdős-Hanani conjecture on asymptotically good designs Frankl-Rödl (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

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

70 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 ( ) P X i µ > ɛµ < e cɛµ. i

71 Setup H u u 1 u 2 x colored W t v H t uncolored U t

72 The Algorithm (k = 3) C = [q] set of colors

73 The Algorithm (k = 3) C = [q] set of colors U t set of currently uncolored vertices

74 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

75 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

76 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

77 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 p t u [0, 1] C, u U t vector of probabilities of colors

78 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 p t u [0, 1] C, u U t vector of probabilities of colors p 0 u = (1/q,..., 1/q) initial color vector

79 H u u 1 u 2 x colored W t v H t uncolored U t

80 For u U, c [q], tentatively activate c at u with probability Θ p u (c).

81 For u U, c [q], tentatively activate c at u with probability A color is lost at u if either Θ p u (c). there is an edge uu 1 u 2 such that c is tentatively activated at u 1 and u 2, or

82 For u U, c [q], tentatively activate c at u with probability A color is lost at u if either Θ p u (c). 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

83 For u U, c [q], tentatively activate c at u with probability A color is lost at u if either Θ p u (c). 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

84 For u U, c [q], tentatively activate c at u with probability A color is lost at u if either Θ p u (c). 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]

85 Parameters (k = 3) During the process, we must choose update values to maintain the values of certain parameters: c p u(c) 1 e uvw = c p u(c)p v (c)p w (c) log ( t deg(v) 1 1 log ) 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.

86 What next? Independence number of locally sparse Graphs Let G contain no K 4

87 What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erdős-Komlós-Szemerédi (1981) α(g) > c n log log

88 What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erdős-Komlós-Szemerédi (1981) α(g) > c n log log Shearer (1995) α(g) > c n log log log

89 What next? Independence number of locally sparse Graphs Let G contain no K 4 Ajtai-Erdős-Komlós-Szemerédi (1981) α(g) > c n log log Shearer (1995) α(g) > c n log log log Major Open Conjecture (Erdős et. al.) α(g) > c n log

90 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 χ(h) < c ( ) 1 k 1. log

91 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 χ(h) < c Weaker Conjecture: χ(h) = o( 1 k 1 ) ( ) 1 k 1. log

92 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 χ(h) < c Weaker Conjecture: χ(h) = o( 1 k 1 ) Algorithms?? ( ) 1 k 1. log 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

93 An Application to Theoretical Computer Science For fixed and n, a simple randomized algorithm yields ( n ) α(h) = Ω 1/(k 1)

94 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. ( ( ) ) log 1/(k 1) α(h) = O n H is 2-colorable

95 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. ( ( ) ) log 1/(k 1) 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

96 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 points in the plane always contain 2n 2 points which do not determine a right angle?

97 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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)

98 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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

99 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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 Erdős (1977) Ω(n 2/3 )

100 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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 Erdős (1977) Ω(n 2/3 ) ( ) n Elekes (2009) Ω log n

101 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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 Erdős (1977) Ω(n 2/3 ) ( ) n Elekes (2009) Ω log n Gyárfás-M Ω(n) assuming Frieze-M Conjecture holds for k = 3 and F = K 3 9

102 An Application to Discrete Geometry Problem (Erdős 1977) Do n 2 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 Erdős (1977) Ω(n 2/3 ) ( ) n Elekes (2009) Ω log n Gyárfás-M Ω(n) assuming Frieze-M Conjecture holds for k = 3 and F = K 3 9 What about in 3-dimensions?

103 Thank You