On Hypergraphs with Every Four Points Spanning at Most Two Triples

Let F be a triple system on an n element set. Suppose that F contains more than (1/3 ) n 3 triples, where >10 6 is explicitly defined and n is suciently large. Then there is a set of four points containing at least three triples of F .T his improves previous bounds of de Caen [1] and Matthias [7] .

[1]  György Turán,et al.  On frequent sets of Boolean matrices , 1998, Annals of Mathematics and Artificial Intelligence.

[2]  G. Andrews,et al.  A new four parameter q-series identity and its partition implications , 2002, math/0205055.

[3]  Vojtech Rödl,et al.  On the Turán Number of Triple Systems , 2002, J. Comb. Theory, Ser. A.

[4]  H. Rosengren.  Karlsson-Minton type hypergeometric functions on the root system Cn , 2002, math/0205108.

[5]  H. Rosengren.  Reduction Formulas for Karlsson–Minton-Type Hypergeometric Functions , 2002, math/0202232.

[6]  S. Warnaar,et al.  Summation and transformation formulas for elliptic hypergeometric series , 2000, math/0001006.

[7]  M. Schlosser.  Multilateral transformations of q-series with quotients of parameters that are nonnegative integer powers of q , 2001, math/0102174.

[8]  G. Andrews.  Bailey’s Transform, Lemma, Chains and Tree , 2001 .

[9]  Frits Beukers,et al.  SPECIAL FUNCTIONS (Encyclopedia of Mathematics and its Applications 71) , 2001 .

[10]  M. Schlosser.  Elementary derivations of identities for bilateral basic hypergeometric series , 2000, math/0010161.

[11]  Michael J. Schlosser,et al.  A new multidimensional matrix inverse with applications to multiple q-series , 1999, Discret. Math..

[12]  Linyuan Lu.  An Upper Bound for the Turr an Number T 3 (n; 4) , 1999 .

[13]  Gaurav Bhatnagar,et al.  Generalized Bibasic Hypergeometric Series and TheirU(n) Extensions , 1997 .

[14]  Christian Krattenthaler,et al.  A new matrix inverse , 1996 .

[15]  M. Ismail,et al.  $q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals , 1994 .

[16]  S. Milne,et al.  TheCℓ Bailey transform and Bailey lemma , 1993 .

[17]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[18]  C. Wenchang.  Some Multifold Reciprocal Transformations with Applications to Series Expansions , 1991 .

[19]  Z. Füredi.  Surveys in Combinatorics, 1991: “Turán Type Problems” , 1991 .

[20]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[21]  Mizan Rahman,et al.  An Indefinite Bibasic Summation Formula and Some Quadratic, Cubic and Quartic Summation and Transformation Formulas , 1990, Canadian Journal of Mathematics.

[22]  George Gasper,et al.  Summation, transformation, and expansion formulas for bibasic series , 1989 .

[23]  George E. Andrews,et al.  q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra , 1986 .

[24]  W. A. Al-Salam,et al.  On Quadratic Transformations of Basic Series , 1984 .

[25]  Zoltán Füredi,et al.  An exact result for 3-graphs , 1984, Discret. Math..

[26]  Miklós Simonovits,et al.  Supersaturated graphs and hypergraphs , 1983, Comb..

[27]  Ira M. Gessel,et al.  Applications of q-lagrange inversion to basic hypergeometric series , 1983 .

[28]  S. Milne.  A multiple series transformation of the very well poised $_{2k+4}\Psi_{2k+4}$. , 1980 .

[29]  Richard Askey,et al.  The very well poised , 1979 .

[30]  George E. Andrews,et al.  Applications of Basic Hypergeometric Functions , 1974 .

[31]  H. W. Gould,et al.  Some new inverse series relations , 1973 .

[32]  L. Carlitz,et al.  Some inverse relations , 1973 .

[33]  H. S. Shukla.  A note on the sums of certain bilateral hypergeometric series , 1959 .

[34]  M. Jackson.  ON WELL-POISED BILATERAL HYPER-GEOMETRIC SERIES OF THE TYPE 8 ψ 8 , 1950 .

[35]  W. N. Bailey.  WELL-POISED BASIC HYPERGEOMETRIC SERIES , 1947 .

[36]  W. N. Bailey.  SERIES OF HYPERGEOMETRIC TYPE WHICH ARE INFINITE IN BOTH DIRECTIONS , 1936 .

[37]  F. Whipple.  A Group of Generalized Hypergeometric Series: Relations Between 120 Allied Series of the Type F(a,b,ce,f) , 1925 .

[38]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[39]  L. J. Rogers.  Second Memoir on the Expansion of certain Infinite Products , 1893 .