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Michael Tait Polarity graphs Sidon sets and C 4 free graphs Orthogonal polarity graphs and Sidon sets Results Open Problems Michael Tait University of California-San Diego mtait@math.ucsd.edu Joint work with Craig Timmons June 5, 2014

  1. Michael Tait Polarity graphs Sidon sets and C 4 free graphs Orthogonal polarity graphs and Sidon sets Results Open Problems Michael Tait University of California-San Diego mtait@math.ucsd.edu Joint work with Craig Timmons June 5, 2014 Michael Tait (UCSD) June 5, 2014 1 / 24

  2. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Let ( P , L , I ) be a set of points , a set of lines , and a set of Results incidences in P × L . Open Problems Michael Tait (UCSD) June 5, 2014 2 / 24

  3. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Let ( P , L , I ) be a set of points , a set of lines , and a set of Results incidences in P × L . Open Problems Definition A polarity is a map π : P ∪ L → P ∪ L with the following properties: Michael Tait (UCSD) June 5, 2014 2 / 24

  4. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Let ( P , L , I ) be a set of points , a set of lines , and a set of Results incidences in P × L . Open Problems Definition A polarity is a map π : P ∪ L → P ∪ L with the following properties: 1 π ( P ) = L and π ( L ) = P . Michael Tait (UCSD) June 5, 2014 2 / 24

  5. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Let ( P , L , I ) be a set of points , a set of lines , and a set of Results incidences in P × L . Open Problems Definition A polarity is a map π : P ∪ L → P ∪ L with the following properties: 1 π ( P ) = L and π ( L ) = P . 2 π 2 = id . Michael Tait (UCSD) June 5, 2014 2 / 24

  6. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Let ( P , L , I ) be a set of points , a set of lines , and a set of Results incidences in P × L . Open Problems Definition A polarity is a map π : P ∪ L → P ∪ L with the following properties: 1 π ( P ) = L and π ( L ) = P . 2 π 2 = id . 3 For p ∈ P and l ∈ L , ( π ( l ) , π ( p )) ∈ I if and only if ( p , l ) ∈ I . Michael Tait (UCSD) June 5, 2014 2 / 24

  7. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Given a geometry ( P , L , I ) and a polarity π , one can Open Problems construct a polarity graph. Michael Tait (UCSD) June 5, 2014 3 / 24

  8. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Given a geometry ( P , L , I ) and a polarity π , one can Open Problems construct a polarity graph. V ( G π ) = P Michael Tait (UCSD) June 5, 2014 3 / 24

  9. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Given a geometry ( P , L , I ) and a polarity π , one can Open Problems construct a polarity graph. V ( G π ) = P E ( G π ) = {{ p , q } : p � = q ∈ P , ( p , π ( q )) ∈ I} . Michael Tait (UCSD) June 5, 2014 3 / 24

  10. Polarities Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Given a geometry ( P , L , I ) and a polarity π , one can Open Problems construct a polarity graph. V ( G π ) = P E ( G π ) = {{ p , q } : p � = q ∈ P , ( p , π ( q )) ∈ I} . If ( p , π ( p )) ∈ I then p is called an absolute point . Michael Tait (UCSD) June 5, 2014 3 / 24

  11. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Let P be the one-dimensional subspaces of F 3 q and L be the Open Problems two-dimensional subspaces. Michael Tait (UCSD) June 5, 2014 4 / 24

  12. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Let P be the one-dimensional subspaces of F 3 q and L be the Open Problems two-dimensional subspaces. Define I by containment. i.e. ( P , L , I ) is a finite projective plane of order q . Michael Tait (UCSD) June 5, 2014 4 / 24

  13. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Let P be the one-dimensional subspaces of F 3 q and L be the Open Problems two-dimensional subspaces. Define I by containment. i.e. ( P , L , I ) is a finite projective plane of order q . Define a map π that sends points and lines to their orthogonal complements. Michael Tait (UCSD) June 5, 2014 4 / 24

  14. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Let P be the one-dimensional subspaces of F 3 q and L be the Open Problems two-dimensional subspaces. Define I by containment. i.e. ( P , L , I ) is a finite projective plane of order q . Define a map π that sends points and lines to their orthogonal complements. π is a polarity. Michael Tait (UCSD) June 5, 2014 4 / 24

  15. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Let P be the one-dimensional subspaces of F 3 q and L be the Open Problems two-dimensional subspaces. Define I by containment. i.e. ( P , L , I ) is a finite projective plane of order q . Define a map π that sends points and lines to their orthogonal complements. π is a polarity. ( x 0 , x 1 , x 2 ) ∼ ( y 0 , y 1 , y 2 ) when x 0 y 0 + x 1 y 1 + x 2 y 2 = 0. Michael Tait (UCSD) June 5, 2014 4 / 24

  16. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs This particular polarity graph was constructed by Erd˝ os, Results R´ enyi, and S´ os (1966) and by Brown (1966) in relation to a Open Problems problem in extremal graph theory. Michael Tait (UCSD) June 5, 2014 5 / 24

  17. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs This particular polarity graph was constructed by Erd˝ os, Results R´ enyi, and S´ os (1966) and by Brown (1966) in relation to a Open Problems problem in extremal graph theory. We will call this graph the Erd˝ os-R´ enyi polarity graph and denote it by ER q . Michael Tait (UCSD) June 5, 2014 5 / 24

  18. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs This particular polarity graph was constructed by Erd˝ os, Results R´ enyi, and S´ os (1966) and by Brown (1966) in relation to a Open Problems problem in extremal graph theory. We will call this graph the Erd˝ os-R´ enyi polarity graph and denote it by ER q . ER q has q 2 + q + 1 vertices and 1 2 q ( q + 1) 2 edges. Michael Tait (UCSD) June 5, 2014 5 / 24

  19. A polarity graph Michael Tait Polarity graphs Sidon sets and C 4 free graphs This particular polarity graph was constructed by Erd˝ os, Results R´ enyi, and S´ os (1966) and by Brown (1966) in relation to a Open Problems problem in extremal graph theory. We will call this graph the Erd˝ os-R´ enyi polarity graph and denote it by ER q . ER q has q 2 + q + 1 vertices and 1 2 q ( q + 1) 2 edges. ER q does not contain C 4 as a subgraph. Michael Tait (UCSD) June 5, 2014 5 / 24

  20. ER 4 Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Open Problems Michael Tait (UCSD) June 5, 2014 6 / 24

  21. ER 8 Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Open Problems Michael Tait (UCSD) June 5, 2014 7 / 24

  22. ER 16 Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Open Problems Michael Tait (UCSD) June 5, 2014 8 / 24

  23. C 4 free graphs Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results The study of C 4 free graphs with many edges has a rich Open Problems history in extremal combinatorics. Michael Tait (UCSD) June 5, 2014 9 / 24

  24. C 4 free graphs Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results The study of C 4 free graphs with many edges has a rich Open Problems history in extremal combinatorics. Definition The Tur´ an number for C 4 is the maximum number of edges in an n -vertex graph that does not contain C 4 as a subgraph. This quantity is denoted by ex ( n , C 4 ). Michael Tait (UCSD) June 5, 2014 9 / 24

  25. C 4 free graphs Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results The study of C 4 free graphs with many edges has a rich Open Problems history in extremal combinatorics. Definition The Tur´ an number for C 4 is the maximum number of edges in an n -vertex graph that does not contain C 4 as a subgraph. This quantity is denoted by ex ( n , C 4 ). ER q implies that ex ( q 2 + q + 1 , C 4 ) ≥ 1 2 q ( q + 1) 2 . Michael Tait (UCSD) June 5, 2014 9 / 24

  26. C 4 free graphs Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Open Problems A counting argument of K˝ ov´ ari, S´ os, and Tur´ an (1954) gives that ex ( n , C 4 ) ≤ 1 2 n 3 / 2 + 1 2 n . Michael Tait (UCSD) June 5, 2014 10 / 24

  27. C 4 free graphs Michael Tait Polarity graphs Sidon sets and C 4 free graphs Results Open Problems A counting argument of K˝ ov´ ari, S´ os, and Tur´ an (1954) gives that ex ( n , C 4 ) ≤ 1 2 n 3 / 2 + 1 2 n . Combined with the lower bound from ER q , ex ( n , C 4 ) ∼ 1 2 n 3 / 2 . Michael Tait (UCSD) June 5, 2014 10 / 24

  28. C 4 free graphs: exact results Michael Tait Computer search gives ex ( n , C 4 ) for n ≤ 31 Polarity graphs Sidon sets and C 4 free graphs Results Open Problems Michael Tait (UCSD) June 5, 2014 11 / 24

  29. C 4 free graphs: exact results Michael Tait Computer search gives ex ( n , C 4 ) for n ≤ 31 Polarity graphs Sidon sets and C 4 free graphs Theorem (F¨ uredi) Results Let q be a prime power. Then Open Problems ex ( q 2 + q + 1 , C 4 ) = 1 2 q ( q + 1) 2 . Furthermore if q is even or if q > 13 , then ER q is the unique extremal graph. Michael Tait (UCSD) June 5, 2014 11 / 24

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