Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs

Three classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size.

[1]  P. Erdős,et al.  COLLOQUIA MATHEMATICA SOCIETATIS JÁNOS BOLYAI 4 . COMBINATORIAL THEORY AND ITS APPLICATIONS , 1969 .

[2]  H. Furstenberg,et al.  A density version of the Hales-Jewett theorem , 1991 .

[3]  T. Apostol.  Introduction to analytic number theory , 1976 .

[4]  J. Singer.  A theorem in finite projective geometry and some applications to number theory , 1938 .

[5]  Ernesto Damiani,et al.  Decompositions of Bn and PIn Using Symmetric Chains , 1994, J. Comb. Theory, Ser. A.

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

[7]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[8]  D. Hilbert,et al.  Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten , 1933 .

[9]  Ronald L. Graham,et al.  On Multicolor Ramsey Numbers for Complete Bipartite Graphs , 1975 .

[10]  M. Aigner.  Combinatorial Order Theory , 1979 .

[11]  Peter Volkmann,et al.  Über ein Problem von Fenyő , 1984 .

[12]  P. Erdos,et al.  On collections of subsets containing no 4-member Boolean algebra. , 1971 .

[13]  Igor Rivin,et al.  On some extremal problems in graph theory , 1999 .

[14]  A. Rényii,et al.  ON A PROBLEM OF GRAPH THEORY , 1966 .

[15]  P. Erdös,et al.  Graph Theory and Probability , 1959 .

[16]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[17]  P. Erdös,et al.  On a problem of sidon in additive number theory, and on some related problems , 1941 .

[18]  Peter Braß,et al.  On the maximum number of edges in a c4-free subgraph of qn , 1995, J. Graph Theory.

[19]  P. Cameron.  Combinatorics: Topics, Techniques, Algorithms , 1995 .

[20]  I. Reiman.  Über ein Problem von K. Zarankiewicz , 1958 .

[21]  Martin Aigner,et al.  Lexicographic matching in Boolean algebras , 1973 .

[22]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[23]  E. Szemeri~di,et al.  On Sets of Integers Containing No Four Elements in Arithmetic Progression , .

[24]  Saharon Shelah,et al.  On problems of Moser and Hanson , 1972 .

[25]  P. Erdös.  On extremal problems of graphs and generalized graphs , 1964 .

[26]  Daniel J. Kleitman,et al.  Partitioning a power set into union-free classes , 1991, Discret. Math..

[27]  Zoltán Füredi,et al.  Ramsey-Sperner theory , 1987, Discret. Math..

[28]  Chung-Tao Yang.  A theorem in finite projective geometry , 1949 .

[29]  Zoltán Füredi,et al.  On the Number of Edges of Quadrilateral-Free Graphs , 1996, J. Comb. Theory, Ser. B.

[30]  L. Beineke,et al.  Selected Topics in Graph Theory 2 , 1985 .

[31]  D. Kleitman,et al.  Proof techniques in the theory of finite sets , 1978 .

[32]  Daniel J. Kleitman,et al.  Strong Versions of Sperner's Theorem , 1976, J. Comb. Theory, Ser. A.

[33]  Zoltán Füredi.  A Ramsey-Sperner theorem , 1985, Graphs Comb..

[34]  D. Hilbert.  Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten. , 1892 .

[35]  Paul Erdös,et al.  Quantitative Forms of a Theorem of Hilbert , 1985, J. Comb. Theory, Ser. A.

[36]  F. Behrend.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Ronald L. Graham,et al.  Rudiments of Ramsey theory , 1981 .

[38]  Vojtech Rödl,et al.  Extremal Problems for Affine Cubes of Integers , 1998, Comb. Probab. Comput..

[39]  D. Lubell.  A Short Proof of Sperner’s Lemma , 1966 .

[40]  W. G. Brown.  On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.

[41]  J. Spencer.  Probabilistic Methods in Combinatorics , 1974 .

[42]  Zoltán Füredi,et al.  Union-Free Families of Sets and Equations over Fields , 1986 .

[43]  Hillel Furstenberg,et al.  Recurrent Ergodic Structures and Ramsey Theory , 1990 .