Combinatorial Dichotomies in Set Theory

We give an overview of a research line concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal. While this is of great interest to set theorists, one of the main motivations behind this line of research is in its applicability to other areas of mathematics. We give some details about this and we expose some possible directions for further research.

[1]  Z. Balogh ON COMPACT HAUSDORFF SPACES OF COUNTABLE TIGHTNESS , 2010 .

[2]  P. Erdos,et al.  On chromatic number of graphs and set-systems , 1966 .

[3]  S. Todorcevic Two examples of Borel partially ordered sets with the countable chain condition , 1991 .

[4]  Alan D. Taylor,et al.  Saturation properties of ideals in generic extensions. I , 1982 .

[5]  Boban Velickovic,et al.  CCC forcing and splitting reals , 2005 .

[6]  S. Argyros,et al.  Unconditional families in Banach spaces , 2008 .

[7]  I. Farah ALL AUTOMORPHISMS OF THE CALKIN ALGEBRA ARE INNER , 2007, 0705.3085.

[8]  James E. Baumgartner,et al.  SATURATION PROPERTIES OF IDEALS IN GENERIC EXTENSIONS. II , 2010 .

[9]  J. Cremona,et al.  Proceedings of the London Mathematical Society , 1893 .

[10]  S. Shelah,et al.  Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II , 1988 .

[11]  R. G. Douglas,et al.  Extensions of C*-algebras and K-homology , 1977 .

[12]  S. Todorcevic,et al.  COFINAL TYPES OF TOPOLOGICAL DIRECTED ORDERS , 2004 .

[13]  William S. Hatcher,et al.  Foundations of Mathematics , 1968 .

[14]  A modification of Shelah’s oracle-c.c. with applications , 1992 .

[15]  H. Rosenthal Point-Wise Compact Subsets of the First Baire Class , 1977 .

[16]  S. Todorcevic Biorthogonal systems and quotient spaces via baire category methods , 2006 .

[17]  Fred Galvin,et al.  On the Singular Cardinals Problem , 1981 .

[18]  Richard Rado,et al.  Theorems on intervals of ordered sets , 1981, Discret. Math..

[19]  Philipp Doebler Rado's Conjecture Implies that All stationary Set Preserving Forcings are Semiproper , 2013, J. Math. Log..

[20]  R. Rado,et al.  Covering Theorems for Ordered Sets , 1948 .

[21]  A. Ostaszewski TOPOLOGY AND BOREL STRUCTURE , 1976 .

[22]  Robert E. Woodrow,et al.  Finite and Infinite Combinatorics in Sets and Logic , 1993 .

[23]  Ilijas Farah,et al.  Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers , 2000 .

[24]  M. Rudin Lectures on Set Theoretic Topology , 1975 .

[25]  Thomas Jech,et al.  Weak Distributivity, A Problem of Von Neumann and the Mystery of Measurability , 2006, Bulletin of Symbolic Logic.

[26]  S. Todorcevic Walks On Ordinals And Their Characteristics , 2007 .

[27]  W. Johnson,et al.  On Banach spaces whose dual balls are not weak∗ sequentially compact , 1977 .

[28]  Between Maharam's and von Neumann's problems , 2004, math/0401134.

[29]  W. Just Nowhere Dense P-Subsets of ω ∗ , 1989 .

[30]  Andreas Blass,et al.  A partition theorem for perfect sets , 1981 .

[31]  Reflection implies the SCH , 2004, math/0404323.

[32]  Stevo Todorcevic Gaps in analytic quotients , 1998 .

[33]  B. Velickovic OCA and automorphisms of P(ω) ⧸ fin , 1993 .

[34]  Z. Szentmiklóssy,et al.  Convergent free sequences in compact spaces , 1992 .

[35]  John W. Tukey,et al.  Convergence and uniformity in topology , 1940 .

[36]  John R. Steel PFA implies ADL(ℝ) , 2005, J. Symb. Log..

[37]  D. Fremlin The partially ordered sets of measure theory and Tukey's ordering , 1991 .

[39]  J. Christensen Some results with relation to the control measure problem , 1978 .

[40]  Stevo Todorcevic,et al.  Oscillations of Sets of Integers , 1998 .

[41]  R. DeVille,et al.  Smoothness and renormings in Banach spaces , 1993 .

[42]  Moti Gitik Nonsplitting subset of κ (κ , 1985 .

[43]  Michael A. Coco Biorthogonal systems in Banach spaces , 2003 .

[44]  Stevo Todorcevic,et al.  Introduction to Ramsey Spaces , 2010 .

[45]  Dag Normann,et al.  Logic Colloquium 2005 , 2007 .

[46]  Stevo Todorcevic Remarks on Martin's axiom and the continuum hypothesis , 1991 .

[47]  Stevo Todorcevic Conjectures of Rado and Chang and Cardinal Arithmetic , 1993 .

[48]  D. Maharam An Algebraic Characterization of Measure Algebras , 1947 .

[49]  Saharon Shelah,et al.  On the consistency of some partition theorems for continuous colorings, and the structure of aleph1-dense real order types , 1985, Ann. Pure Appl. Log..

[50]  Stevo Todorcevic,et al.  Introduction to Ramsey Spaces (AM-174) , 2010 .

[51]  Stevo Todorcevic,et al.  Partition Problems In Topology , 1989 .

[52]  S. Mazur Über schwache Konvergenz in den Raümen ($L^{p}$) , 1933 .

[53]  S. Todorcevic,et al.  Directed sets and cofinal types , 1985 .

[54]  A. Tarski,et al.  On Families of Mutually Exclusive Sets , 1943 .

[55]  M. Talagrand,et al.  Pointwise Compact Sets of Baire-Measurable Functions , 1978 .

[56]  J. Mujica Separable quotients of Banach spaces , 1997 .

[57]  Justin Tatch Moore,et al.  Open colorings, the continuum and the second uncountable cardinal , 2002 .

[58]  Max L. Warshauer,et al.  Lecture Notes in Mathematics , 2001 .

[59]  W. Woodin,et al.  Supercompact cardinals, sets of reals, and weakly homogeneous trees. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[60]  P. Dodos,et al.  On pairs of definable orthogonal families , 2008, 0805.2033.

[61]  Martin's axiom and partitions , 1987 .

[62]  Robert M. Solovay,et al.  Real-valued measurable cardinals , 1967 .

[63]  F. Hausdorff,et al.  Summen von $ℵ_1$ Mengen , 1936 .

[64]  S. Todorcevic,et al.  Forcing positive partition relations , 1983 .

[65]  M. Talagrand Maharam’s problem , 2006, math/0601689.

[66]  W. Just Repercussions of a Problem of Erdős and Ulam on Density Ideals , 1990, Canadian Journal of Mathematics.

[67]  Justin Tatch Moore A solution to the L space problem , 2005 .

[68]  R. Jensen,et al.  The fine structure of the constructible hierarchy , 1972 .

[69]  M. Foreman An ℵ1-dense ideal on ℵ2 , 1998 .

[70]  Slawomir Solecki,et al.  Analytic Ideals and Their Applications , 1999, Ann. Pure Appl. Log..

[71]  Stevo Todorcevic,et al.  Comparing the Continuum with the First Two Uncountable Cardinals , 1997 .

[72]  Uri Abraham,et al.  Partition properties of ω1 compatible with CH , 1997 .

[73]  Qi Feng,et al.  Homogeneity for open partitions of pairs of reals , 1993 .

[74]  Countable spread of exp Y and λY , 1982 .

[75]  Mohamed Bekkali Topics in Set Theory: Lebesgue Measurability, Large Cardinals, Forcing Axioms, Rho-functions , 1991 .

[76]  S. Todorcevic Compact subsets of the first Baire class , 1999 .

[77]  B. Velickovic Definable automorphisms of () , 1986 .

[78]  Moti Gitik Nonsplitting Subset of P (+) , 1985, J. Symb. Log..

[79]  Ernest Schimmerling,et al.  Characterization of □κ in Core Models , 2004, J. Math. Log..

[80]  J. P. Moreno,et al.  Renorming Banach spaces with the Mazur intersection property , 1997 .

[81]  W. Hugh Woodin,et al.  The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal , 1999 .

[82]  S. Todorcevic,et al.  Generic Banach spaces and generic simplexes , 2011 .

[83]  The sequential topology on complete Boolean algebras , 1996, math/9612207.

[84]  J. Hirschorn Random trees under CH , 2007 .

[85]  W. Johnson,et al.  On ω*-basic sequences and their applications to the study of Banach spaces , 1972 .

[86]  W. Johnson,et al.  On the existence of fundamental and total founded biorthogonal systems in Banach spaces , 1973 .

[87]  S. Todorcevic,et al.  Analytic k-spaces , 2005 .

[88]  S. Todorcevic On a Conjecture of R. Rado , 1983 .

[89]  Saharon Shelah,et al.  On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ 1 -Dense Real Order Types , 2002 .

[90]  Π2 CONSEQUENCES OF BMM + NSω1 IS PRECIPITOUS AND THE SEMIPROPERNESS OF STATIONARY SET PRESERVING FORCINGS , 2009 .

[91]  Stevo Todorcevic A dichotomy for P-ideals of countable sets , 2000 .

[92]  S. Todorcevic A problem of von Neumann and Maharam about algebras supporting continuous submeasures , 2004 .

[93]  Ralf Schindler,et al.  Stacking mice , 2009, The Journal of Symbolic Logic.

[94]  J R Steel,et al.  Projective determinacy. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[95]  Stevo Todorcevic,et al.  BOREL CHROMATIC NUMBERS , 1999 .

[96]  S. Todorcevic,et al.  Banach spaces and Ramsey Theory: some open problems , 2010, 1006.2668.

[97]  Boban Velickovic,et al.  Definable Automorphisms of P(ω)/fin , 1986 .

[98]  S. Todorcevic Some Applications of S and L Combinatorics a , 1993 .

[99]  T. Jech,et al.  Complete CCC Boolean Algebras, the Order Sequential Topology, and a Problem of Von Neumann , 2003, math/0312473.

[100]  Kenneth Kunen,et al.  Set Theory: An Introduction to Independence Proofs , 2010 .

[101]  Stevo Todorcevic Oscillations of Real Numbers , 1987 .

[102]  P. Koszmider On a problem of Rolewicz about Banach spaces that admit support sets , 2009 .

[104]  H. Rosenthal,et al.  A double-dual characterization of separable Banach spaces containingl1 , 1975 .

[105]  Qi Feng Rado's Conjecture and Presaturation of The Nonstationary Ideal on Omega1 , 1999, J. Symb. Log..

[106]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[107]  A Weak Version of at from OCA , 1992 .

[108]  A. Leiderman,et al.  Final compactness is not preserved under multiplication of spaces of the form Cp(X) , 1988 .

[109]  J. van Mill,et al.  Open Problems in Topology , 1990 .

[110]  N. Weaver,et al.  The Calkin algebra has outer automorphisms , 2006, math/0606594.

[111]  J. Silver,et al.  Counting the number of equivalence classes of Borel and coanalytic equivalence relations , 1980 .

[112]  P. Hájek,et al.  Mazur intersection property for Asplund spaces , 2008, 0804.0583.

[113]  Multiple gaps , 2010, 1001.4888.

[114]  F. Drake CONSEQUENCES OF MARTIN'S AXIOM , 1985 .

[115]  Review: D. H. Fremlin, Consequences of Martin's Axiom , 1988 .

[116]  R. Daniel Mauldin,et al.  The Scottish book : mathematics from the Scottish Café , 1981 .

[117]  S. Ziemiański,et al.  On the Occasion of his Seventy Fifth Birthday , 2006 .

[118]  Keith Devlin PROPER FORCING(Lecture Notes in Mathematics, 940) , 1983 .