No CrossRef data available.
Article contents
THE ERDŐS–HAJNAL PROBLEM LIST
Part of:
Set theory
Published online by Cambridge University Press: 06 January 2025
Abstract
We give an update on the problem list of Erdős and Hajnal.
Keywords
MSC classification
Primary:
03E02: Partition relations
03E05: Other combinatorial set theory
03E99: None of the above, but in this section
Secondary:
03E35: Consistency and independence results
Information
- Type
- Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
Footnotes
Dedicated to Pál Erdős (1913–1996) and András Hajnal (1931–2016)
References
Abraham, U.,
Free sets for nowhere-dense set mappings
.
Israel Journal of Mathematics
, vol. 39 (1981), pp. 167–176.Google Scholar
Abraham, U. and Magidor, M.,
Cardinal arithmetic
,
Handbook of Set Theory
(M. Foreman and A. Kanamori, editors), II, Springer, 2010, pp. 1149–1227.Google Scholar
Bagemihl, F.,
The existence of an everywhere dense independent set
.
Michigan Math. Journal
, vol. 20 (1973), pp. 1–2.Google Scholar
Bagemihl, F.,
Independent sets and second category
.
Order
, vol. 7 (1990), pp. 65–66.Google Scholar
Baumgartner, J. E.,
$\mathrm{ADT}\Rightarrow \mathrm{KH}$
, unpublished notes, cca, 1971.Google Scholar
$\mathrm{ADT}\Rightarrow \mathrm{KH}$
, unpublished notes, cca, 1971.Google ScholarBaumgartner, J. E.,
Partition relations for uncountable cardinals
.
Israel Journal of Mathematics
, vol. 21 (1975), pp. 296–307.Google Scholar
Baumgartner, J. E.,
Almost-disjoint sets, the dense set problem and the partition calculus
.
Annals of Mathematical Logic
, vol. 10 (1976), pp. 401–439.Google Scholar
Baumgartner, J. E.,
A new class of order types
.
Annals of Mathematical Logic
, vol. 9 (1976), pp. 187–222.Google Scholar
Baumgartner, J. E.,
Generic graph construction
.
Journal of Symbolic Logic
, vol. 49 (1984), pp. 234–240.Google Scholar
Baumgartner, J. E. and Hajnal, A.,
A proof (involving Martin’s axiom) of a partition relation
.
Fundamenta Mathematicæ
, vol. 78, (1973), pp. 193–203.Google Scholar
Baumgartner, J. E. and Hajnal, A., A remark on partition relations for infinite ordinals with an application to finite combinatorics,
Logic and combinatorics (Arcata, Calif., 1985)
(S. G. Simpson, editor), Contemporary Mathematics, 65, American Mathematical Society, Providence, RI, 1987, pp. 157–167.Google Scholar
Baumgartner, J. E., Hajnal, A., and Todorčević, S.,
Extensions of the Erdős-Rado theorem
,
Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991)
, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 411, Kluwer Academic Publishers, Dordrecht, 1993, pp. 1–17.Google Scholar
Baumgartner, R.,
Laver: Iterated perfect set forcing
,
Annals of Mathematical Logic
, vol. 17 (1979), pp. 271–288.Google Scholar
Benda, M. and Ketonen, J.,
Regularity of ultrafilters
.
Israel Journal of Mathematics
, vol. 17 (1974), pp. 231–240.Google Scholar
Ben-David, Sh. and Magidor, M., The weak
$\square$
is really weaker than the full
$\square$
.
The Journal of Symbolic Logic
, vol. 51 (1986), pp. 1029–1033.Google Scholar
$\square$
is really weaker than the full
$\square$
.
The Journal of Symbolic Logic
, vol. 51 (1986), pp. 1029–1033.Google ScholarBen-Neria, O., Garti, S., and Hayut, Y.,
Weak prediction principles
.
Fundamenta Mathematicæ
, vol. 245 (2019), pp. 109–125.Google Scholar
Bergfalk, J.,
Michael Hrusák, Saharon Shelah: Ramsey theory for highly connected monochromatic subgraphs
.
Acta Mathematica Hungarica
, vol. 163 (2021), pp. 309–322.Google Scholar
Blass, A.,
Weak partition relations
.
Proceedings of the American Mathematical Society
, vol. 35 (1972), pp. 249–253.Google Scholar
Chang, C. C.,
A partition theorem for the complete graph on
${\omega}^{\omega }$
.
Journal of Combinatorial Theory
, vol. 12 (1972), pp. 396–452.Google Scholar
${\omega}^{\omega }$
.
Journal of Combinatorial Theory
, vol. 12 (1972), pp. 396–452.Google ScholarDarby, C., Larson, J. A.,
Multicolored graphs on countable ordinals of finite exponent
,
Set Theory (Piscataway, NJ, 1999)
, DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, 58, American Mathematical Society, Providence, RI, 2002, pp. 31–43.Google Scholar
Deiser, O. and Donder, D., Canonical functions, non-regular ultrafilters and Ulam’s problem on
${\unicode{x3c9}}_1$
.
The Journal of Symbolic Logic
, vol. 68 (2003), pp. 713–739.Google Scholar
${\unicode{x3c9}}_1$
.
The Journal of Symbolic Logic
, vol. 68 (2003), pp. 713–739.Google ScholarDevlin, D. C.,
Some partition theorems and ultrafilters on
$\boldsymbol{\omega}$
, Ph. D. thesis, Dartmouth College, July 1979.Google Scholar
$\boldsymbol{\omega}$
, Ph. D. thesis, Dartmouth College, July 1979.Google ScholarDevlin, K. J.,
Some weak versions of large cardinal axioms
.
Annals of Mathematical Logic
, vol. 5 (1972–1973), pp. 291–325.Google Scholar
Devlin, K. J.,
A note on a problem of Erdős and Hajnal
.
Discrete Mathematics
, vol. 11 (1975), pp. 9–22.Google Scholar
Devlin, K. J. and Paris, J., More on the free subset problem.
Annals of Mathematical Logic
, vol. 5 (1972–1973), pp. 327–336.Google Scholar
Donder, H.-D., Jensen, R. B., and Koppelberg, B. J.,
Some applications of the core model
,
Set Theory and Model Theory, Proc. Bonn, 1979
(R. B. Jensen and A. Prestel, editors), Lecture Notes in Mathematics, 872, Springer, Berlin, 1981, pp. 55–97.Google Scholar
Dudek, A. and Rödl, V.,
On the Folkman number
$f\left(2,3,4\right)$
.
Experimental Mathematics
, vol. 17 (2008), pp. 63–67.Google Scholar
$f\left(2,3,4\right)$
.
Experimental Mathematics
, vol. 17 (2008), pp. 63–67.Google ScholarElekes, G.,
Colouring infinite subsets of
$\omega$
,
Colloquia Mathematica Societatis János Bolyai, Infinite and Finite Sets
(A. Hajnal, R. Rado, V. T. Sós, editors), vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 393–396.Google Scholar
$\omega$
,
Colloquia Mathematica Societatis János Bolyai, Infinite and Finite Sets
(A. Hajnal, R. Rado, V. T. Sós, editors), vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 393–396.Google ScholarElekes, G.,
On a partition property of infinite subset of a set
.
Periodica Mathematica Hungarica
, vol. 5 (1974), pp. 215–218.Google Scholar
Elekes, G., Erd, P., and Hajnal, A.,
On some partition properties of families of sets
.
Studia Scientiarum Mathematicarum Hungarica
, vol. 13 (1978), pp. 151–155.Google Scholar
Elekes, G., Hajnal, A., Komjáth, P.,
Partition theorems for the power set
,
Sets, Graphs and Numbers (Budapest, 1991)
(G. Halász, L. Lovász, D. Miklós, and T. Szőnyi, editors), Colloquia Mathematica Societatis János Bolyai, 60, North-Holland, Amsterdam, 1992, pp. 211–217.Google Scholar
Erdős, P.,
Some remarks on set theory
.
Proceedings of the American Mathematical Society
, vol. 1 (1950), pp. 127–141.Google Scholar
Erdős, P.,
An interpolation problem associated with the continuum hypothesis
.
Michigan Mathematical Journal
, vol. 11 (1964), pp. 9–10.Google Scholar
Erdős, P.,
Problems and results in chromatic graph theory
,
Proof Techniques in Graph Theory: Proceedings of the Second Ann Arbor Graph Theory Conference
, Academic Press, Ann Arbor, MI, 1968, 1969, pp. 27–35.Google Scholar
Erdős, P. and Hajnal, A.,
On the structure of set mappings
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 9 (1958), pp. 111–131.Google Scholar
Erdős, P. and Hajnal, A.,
On complete topological subgraphs of certain graphs
.
Annales Universitatis Scientiarum Budapestinensis
, vol. 7 (1964), pp. 143–149.Google Scholar
Erdős, P. and Hajnal, A.,
Some remarks on set theory, IX
.
Michigan Mathematical Journal
, vol. 11 (1964), pp. 107–127.Google Scholar
Erdős, P. and Hajnal, A.,
On chromatic number of graphs and set systems
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 17 (1966), pp. 61–99.Google Scholar
Erdős, P. and Hajnal, A.,
On the chromatic number of infinite graphs
,
Graph Theory Symposium
, János Bolyai Mathematical Society, 1968 Tihany, 1966, pp. 83–98.Google Scholar
Erdős, P. and Hajnal, A.,
Some results and problems for certain polarized partitions
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 21 (1970), pp. 369–392.Google Scholar
Erdős, P. and Hajnal, A.,
Unsolved problems in set theory
,
Proceedings of Symposia in Pure Mathematics
, vol. 13, Providence, RI, 1971, pp. 17–48.Google Scholar
Erdős, P. and Hajnal, A., Unsolved and solved problems in set theory,
Proceedings of the Tarski Symposium, (Proc. Sympos. Pure Math.,
vol. 25, Univ. California, Berkeley, Calif., 1971
, American Mathematical Society, Providence, RI, 1974, pp. 269–287.Google Scholar
Erdős, P. and Hajnal, A.,
Some remarks on set theory XI, Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, III
.
Fundamenta Mathematicae
, vol. 81 (1974), pp. 261–265.Google Scholar
Erdős, P. and Hajnal, A.,
Embedding theorems for graphs establishing negative partition relations
.
Periodica Mathematica Hungarica
, vol. 9 (1978), pp. 205–230.Google Scholar
Erdős, P., Hajnal, A., Máté, A., and Rado, R.,
Combinatorial Set Theory: Partition Relation for Cardinals
, North-Holland, Amsterdam, 1984.Google Scholar
Erdős, P., Hajnal, A., and Milner, E. C.,
On the complete subgraphs of graphs defined by systems of sets
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 17 (1966), pp. 159–229.Google Scholar
Erdős, P., Hajnal, A., and Milner, E. C.,
On sets of almost disjoint subsets of a set
.
Mathematica Academiae Scientiarum Hungaricae
, vol. 19 (1968), pp. 209–218.Google Scholar
Erdős, P., Hajnal, A., and Rado, R.,
Partition relations for cardinal numbers
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 16 (1965), pp. 93–196.Google Scholar
Erdős, P., Hajnal, A., and Shelah, S.,
On some general properties of chromatic numbers
,
Topics In topology(Proc. Colloq., Keszthely, 1972)
(Á. Császár, editor), Colloquia Mathematica Societatis János Bolyai, 8, North-Holland, Amsterdam, 1974, pp. 243–255.Google Scholar
Erdős, P. and Makkai, M.,
Some remarks on set theory, X
.
Studia Scientiarum Mathematicarum Hungarica
, vol. 1 (1966), pp. 157–159.Google Scholar
Erdős, P., Milner, E. C., and Rado, R.,
Intersection theorems for systems of sets, III, Collection of articles dedicated to the memory of Hanna Neumann, IX
.
Journal of the Australian Mathematical Society
, vol. 18 (1974), pp. 22–40.Google Scholar
Erdős, P. and Rado, R.,
A partition calculus in set theory
.
Bulletin of the American Mathematical Society
, vol. 62 (1956), pp. 427–489.Google Scholar
Fleissner, W. G.,
Some spaces related to topological inequalities proven by the Erdős–Rado theorem
.
Proceedings of the American Mathematical Society
, vol. 71 (1978), pp. 313–320.Google Scholar
Folkman, J.,
Graphs with monochromatic complete subgraphs in every edge coloring
.
SIAM Journal on Applied Mathematics
, vol. 18 (1970), pp. 19–24.Google Scholar
Foreman, M.,
Ideals and generic elementary embeddings
,
Handbook of Set Theory
, (M. Foreman and A. Kanamori, editors), Springer, Dordrecht, Heidelberg, London, New York, 2010, pp. 885–1147.Google Scholar
Foreman, M. and Hajnal, A.,
A partition relation for successors of large cardinals
.
Mathematische Annalen
, vol. 325 (2003), pp. 583–623.Google Scholar
Foreman, M. and Laver, R.,
Some downward transfer properties of
${\aleph}_2$
.
Advances in Mathematics
, vol. 67 (1988), pp. 230–238.Google Scholar
${\aleph}_2$
.
Advances in Mathematics
, vol. 67 (1988), pp. 230–238.Google ScholarForeman, M., Magidor, M., and Shelah, S.,
Martin’s maximum, saturated ideals, and nonregular ultrafilters
.
I, Annals of Mathematics
, vol. 127 (1988), pp. 1–47.Google Scholar
Foreman, M., Magidor, M., and Shelah, S.,
Martin’s maximum, saturated ideals, and nonregular ultrafilters
.
II, Annals of Mathematics
, vol. 127 (1988), pp. 521–545.Google Scholar
Fox, J., Grinshpun, A., and Pach, J.,
The Erdős–Hajnal conjecture for rainbow triangles
.
Journal of Combinatorial Theory, (B)
, vol. 111 (2015), pp. 75–125.Google Scholar
Fremlin, D. H.,
Consequences of Martin’s Axiom
, Cambridge University Press, Cambridge, 1984.Google Scholar
Fremlin, D. H.,
Real-valued measurable cardinals
,
Set Theory of the Reals
, (H. Judah, editor), volume 6 of Israel Mathematical Conference Proceedings, Emmy Noether Research Institute for Mathematics (ENI), Bar-Ilan University, Ramat Gan, Ramat Gan, Israel, 1993, 151–304.Google Scholar
Galvin, F.,
On a partition theorem of Baumgartner and Hajnal, colloquia mathematica societatis János Bolyai
,
Infinite and Finite Sets
, vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 711–729.Google Scholar
Galvin, F. and Larson, J.,
Pinning countable ordinals. Collection of articles dedicated to Andrzej Mostowski on his sixtieth birthday, VIII
.
Fundamenta Mathematicæ
, vol. 82 (1974/75), pp. 357–361.Google Scholar
Galvin, F. and Prikry, K.,
Borel sets and Ramsey’s theorem
.
Journal of Symbolic Logic
, vol. 38 (1973), pp. 193–198.Google Scholar
Galvin, F. and Shelah, S.,
Some counterexamples in partition calculus
.
Journal of Combinatorial Theory, (A)
, vol. 15 (1973), pp. 167–174.Google Scholar
Garti, S.,
Unbalanced polarized relations
.
Proceedings of the American Mathematical Society
, vol. 149 (2021), pp. 1281–1287.Google Scholar
Golshani, M.,
(Weak) diamond can fail at the least inaccessible cardinal
.
Fundamenta Mathematicæ
, vol. 256 (2022), pp. 113–129.Google Scholar
Golshani, M.,
Generalized Kurepa hypothesis at singular cardinals
.
Periodica Mathematica Hungarica
, vol. 78 (2019), pp. 200–202.Google Scholar
Gregory, J.,
Souslin trees and the generalized continuum hypothesis
.
Journal of Symbolic Logic
, vol. 41 (1976), pp. 663–671.Google Scholar
Gyárfás, A., Komlós, J., and Szemerédi, E.,
On the distribution of cycle lengths in graphs
.
Journal of Graph Theory
, vol. 8 (1984), pp. 441–462.Google Scholar
Haddad, L. and Sabbagh, G.,
Sur une extension des nombres de Ramsey aux ordinaux
.
Comptes Rendus Math´ematique. Acad´emie des Sciences
, vol. 268 (1969), pp. A1165–A1167.Google Scholar
Hajnal, A.,
Some results and problems in set theory
.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 11 (1960), pp. 277–298.Google Scholar
Hajnal, A.,
Ulam matrices for inaccessible cardinals
.
Bulletin L’Académie Polonaise des Science
, vol. 17 (1969), pp. 683–688.Google Scholar
Hajnal, A.,
Report on the Miklós Schweitzer memorial contest in 1968
.
Matematikai Lapok
, vol. 20 (1969), pp. 145–171 (in Hungarian).Google Scholar
Hajnal, A.,
A negative partition relation
.
Proceedings of the National Academy of Sciences
, vol. 68 (1971), pp. 142–144.Google Scholar
Hajnal, A.,
Paul Erdős’ set theory
,
The mathematics of Paul Erdős, Algorithms Combinatorics II
(R. Graham and J. Nešetřil, editors), vol. 14, Springer, Berlin, 1997, pp. 352–393.Google Scholar
Hajnal, A.,
Rainbow Ramsey theorems for colorings establishing negative partition relations
.
Fundamenta Mathematicæ
, vol. 198 (2008), pp. 255–262.Google Scholar
Hajnal, A. and Juhász, I.,
On discrete subspaces of topological spaces
.
Indagationes Mathematicae
, vol. 29 (1967), pp. 343–356.Google Scholar
Hajnal, A. and Juhász, I.,
Two consistency results in topology
.
Bulletin of the American Mathematical Society
, vol. 73 (1972), pp. 711.Google Scholar
Hajnal, A. and Juhász, I., A consistency result concerning hereditarily
$\alpha$
-Lindelöf spaces.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 24 (1973), pp. 307–312.Google Scholar
$\alpha$
-Lindelöf spaces.
Acta Mathematica Academiae Scientiarum Hungaricae
, vol. 24 (1973), pp. 307–312.Google ScholarHajnal, A., Juhász, I., and Shelah, S.,
Splitting strongly almost disjoint families
.
Transactions of the American Mathematical Society
, vol. 295 (1986), pp. 369–387.Google Scholar
Hajnal, A., Juhász, I., and Shelah, S.,
Strongly almost disjoint families, revisited
.
Fundamenta Mathematicæ
, vol. 163 (2000), pp. 13–23.Google Scholar
Hajnal, A. and Máté, A.,
Set mappings, partitions, and chromatic numbers
,
Logic Colloquium ’73, Bristol, 1973
(J. Shepherdson and H. E. Rose, editors), Studies in Logic and the Foundations of Mathematics, 80, North-Holland, Amsterdam, 1975, pp. 347–379.Google Scholar
Hechler, S. H.,
On two problems in combinatorial set theory
.
Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques
, vol. 20 (1972), 4299–431.Google Scholar
Inamdar, T. and Rinot, A.,
Was Ulam right? I. Basic theory and subnormal ideals
.
Topology and Its Applications
, vol. 323 (2023), Paper No. 108287, 53 pp.Google Scholar
Jech, T.,
Set Theory: The Third Millenium Edition, Revised and Expanded
, Springer Monographs in Mathematics, Springer, Heidelberg, London, New York, 2003.Google Scholar
Jech, T. and Shelah, S.,
A note on canonical functions
.
Israel Journal of Mathematics
, vol. 68 (1989), pp. 376–380.Google Scholar
Jones, A. L.,
A short proof of a partition relation for triples
.
Electronic Journal of Combinatorics
, vol. 7 (2000), res. paper 24, 9 pp.Google Scholar
Jones, A. L.,
A polarized partition relation for weakly compact cardinals using elementary substructures
.
Journal of Symbolic Logic
, vol. 71 (2006), pp. 1342–1352.Google Scholar
Jones, A. L.,
On a result of Szemerédi
.
Journal of Symbolic Logic
, vol. 73 (2008), pp. 953–956.Google Scholar
A. L., Jones,
Even more on partitioning triples of countable ordinals
.
Proceedings of the American Mathematical Society
, vol. 146 (2018), pp. 3529–3539.Google Scholar
Juhász, I. and Shelah, S.,
How large a hereditarily separable or hereditarily Lindelöf space can be ?
Israel Journal of Mathematics
, vol. 53 (1986), pp. 355–364.Google Scholar
Kanamori, A,
On Silver’s and related principles
,
Logic Colloquium ’80
(D. Van Dalen, D. Lascar, and T. J. Smiley, editors), Studies in Logic and the Foundations of Mathematics, 108, North-Holland, Amsterdam, 1982, pp. 153–172.Google Scholar
Kanamori, A.,
Partition relations for successors of cardinals
.
Advances in Mathematics
, vol. 59 (1986), pp. 152–169.Google Scholar
Kanamori, A.,
Erdős and set theory
.
Bulletin of the Symbolic Logic
, vol. 20 (2014), pp. 449–490.Google Scholar
Keisler, H. J.,
A survey of ultraproducts
,
Logic, Methodology and Philosophy of Science, Proc. of the 1964 Congress
, (Y. Bar-Hillel, editor), North-Holland, Amsterdam, 1965, pp. 112–126.Google Scholar
Ketonen, J.,
Nonregular ultrafilters and large cardinals
.
Transactions of the American Mathematical Society
, vol. 224 (1976), pp. 61–73.Google Scholar
Koepke, P.,
The consistency strength of the free–subset property for
${\omega}_{\omega }$
.
Journal of Symbolic Logic
, vol. 49 (1984), pp. 1198–1204.Google Scholar
${\omega}_{\omega }$
.
Journal of Symbolic Logic
, vol. 49 (1984), pp. 1198–1204.Google ScholarKomjáth, P.,
The colouring number
.
Journal of the London Mathematical Society
, vol. 54 (1987), pp. 1–14.Google Scholar
Komjáth, P.,
Erdős’s work on infinite graphs
,
Erdős Centennial
, (L. Lovász, I. Ruzsa, and V. T. Sós, editors), Springer, Berlin, London, New York, 2013, pp. 325–345.Google Scholar
Komjáth, P.,
Notes on some Erdős–Hajnal problems
.
Journal of Symbolic Logic
, vol. 86 (2021), pp. 1116–1123.Google Scholar
Komjáth, P., A comment on a problem of Erdős, Hajnal, and Milner, manuscript.Google Scholar
Komjáth, P. and Pach, J.,
Universal graphs without large bipartite subgraphs
.
Mathematika
, vol. 31 (1984), pp. 282–290.Google Scholar
Komjáth, P. and Shelah, S.,
Forcing constructions for uncountably chromatic graphs
.
Journal of Symbolic Logic
, vol. 53 (1988), pp. 696–707.Google Scholar
Komjáth, P. and Shelah, S.,
A consistent edge partition theorem for infinite graphs
.
Acta Mathematica Hungarica
, vol. 61 (1993), pp. 115–120.Google Scholar
Komjáth, P. and Shelah, S.,
Two consistency results on set mappings
.
Journal of Symbolic Logic
, vol. 65 (2000), pp. 333–338.Google Scholar
Komjáth, P. and Shelah, S.,
Monocolored topological complete graphs in colorings of uncountable complete graphs
.
Acta Mathematica Hungarica
, vol. 163 (2021), pp. 71–84.Google Scholar
Komjáth, P. and Totik, V.,
Problems and Theorems in Classical Set Theory
, Springer, Heidelberg, London, New York, 2006.Google Scholar
Kumar, A. and Shelah, S.,
On a question about families of entire functions
.
Fundamenta Mathematicæ
, vol. 239 (2017), pp. 279–288.Google Scholar
Kunen, K.,
Inacessibility porperties of cardinals
. Ph. D. thesis, Stanford University, 1968.Google Scholar
Lambie-Hanson, C. and Rinot, A.,
Reflections on the coloring and chromatic numbers
.
Combinatorica
, vol. 39 (2019), pp. 165–214.Google Scholar
Lange, A. R., Radziszowski, S. P., and Xu, X., Use of MAX-CUT for Ramsey arrowing triangles.
Journal of Combinatorial Mathematics and Combinatorial Computing
, vol. 88 (2014), pp. 51–71.Google Scholar
Larson, J.,
On some arrow relations
, Ph. D. Thesis, Dartmouth College, Hanover, NH, 1972.Google Scholar
Larson, J. A.,
A short proof of a partition theorem for the ordinal
${\omega}^{\omega }$
.
Annals of Mathematical Logic
, vol. 6 (1973/74), pp. 129–145.Google Scholar
${\omega}^{\omega }$
.
Annals of Mathematical Logic
, vol. 6 (1973/74), pp. 129–145.Google ScholarPaul Larson, S.,
Shelah: Bounding by canonical functions, with CH
.
Journal of Mathematical Logic
, vol. 3 (2003), pp. 193–215.Google Scholar
Laver, R.,
An order type decomposition theorem
.
Annals of Mathematics, (2)
, vol. 98 (1973), pp. 96–119.Google Scholar
Laver, R., Partition relations for uncountable cardinals
$\le {2}^{\aleph_0}$
, colloquia mathematica societatis jános bolyai,
Infinite and Finite Sets
, vol 10, (A. Hajnal, R. Rado, and V. T. Sós, editors), János Bolyai Mathematical Society, Keszthely, 1973, pp. 1029–1042.Google Scholar
$\le {2}^{\aleph_0}$
, colloquia mathematica societatis jános bolyai,
Infinite and Finite Sets
, vol 10, (A. Hajnal, R. Rado, and V. T. Sós, editors), János Bolyai Mathematical Society, Keszthely, 1973, pp. 1029–1042.Google ScholarLaver, R.,
An
$\left({\mathrm{\aleph}}_2,{\mathrm{\aleph}}_2,{\mathrm{\aleph}}_0\right)$
-saturated ideal on
${\unicode{x3c9}}_1$
,
Logic Colloquium ’80
, Studies in Logic and the Foundations of Mathematics, 108 (D. Van Dalen, D. Lascar, and T. J. Smiley, editors), North-Holland, Amsterdam, 1982, pp. 173–180.Google Scholar
$\left({\mathrm{\aleph}}_2,{\mathrm{\aleph}}_2,{\mathrm{\aleph}}_0\right)$
-saturated ideal on
${\unicode{x3c9}}_1$
,
Logic Colloquium ’80
, Studies in Logic and the Foundations of Mathematics, 108 (D. Van Dalen, D. Lascar, and T. J. Smiley, editors), North-Holland, Amsterdam, 1982, pp. 173–180.Google ScholarLiu, H. and Montgomery, R., A solution to Erdős and Hajnal’s odd cycle problem.
Journal of the American Mathematical Society
, vol. 36 (2023), pp. 1131–1234.Google Scholar
Magidor, M.,
On the existence of nonregular ultrafilters and the cardinality of ultrafilters
.
Transactions of the American Mathematical Society
, vol. 249 (1979), pp. 97–111.Google Scholar
Magidor, M. and Shelah, S.,
When does almost free imply free? (For groups, transversals, etc.)
.
Journal of the American Mathematical Society
, vol. 7 (1994), pp. 769–830.Google Scholar
Mekler, A. H. and Shelah, S.,
The consistency strength of “every stationary set reflects”
.
Israel Journal of Mathematics
, vol. 67 (1989), pp. 353–366.Google Scholar
Miller, E. W.,
On a property of families of sets
.
Comptes Rendus Varsovie
, vol. 30 (1937), pp. 31–38.Google Scholar
Mills, C. and Prikry, K., Some recent results about partitions of
${\omega}_1$
, unpublished.Google Scholar
${\omega}_1$
, unpublished.Google ScholarMilner, E. C.,
Partition relations for ordinal numbers
.
Canadian Journal of Mathematics
, vol. 21 (1969), pp. 317–334.Google Scholar
Milner, E. C. and Prikry, K.,
A partition theorem for triples
.
Proceedings of the American Mathematical Society
, vol. 97 (1986), pp. 488–494.Google Scholar
Milner, E. C. and Prikry, K.,
A partition relation for triples using a model of Todorčević, Directions in infinite graph theory and combinatorics (Cambridge
.
Discrete Mathematics
, vol. 95 (1989), no. 1991
, pp. 183–191.Google Scholar
Milner, E. C. and Rado, R.,
The pigeonhole principle for ordinal numbers
.
Journal of the London Mathematical Society
, vol. 15 (1965), pp. 750–768.Google Scholar
Milner, E. C. and Shelah, S.,
Some theorems on transversals, Colloquia Mathematica Societatis János Bolyai
,
Infinite and Finite Sets
(A. Hajnal, R. Rado, and V. T. Sós, editors), vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 1115–1126.Google Scholar
Miyamoto, T., Separating a family of weak Kurepa hypotheses and the transversal hypothesis. 2005, http://www.seto.nanzan-u.ac.jp/st/nas/academia/vol005.pdf/05-025-038.pdf
Google Scholar
Nesteril, J. and Rödl, V.,
A Ramsey graph without triangles exists for any graph without triangles
,
Colloquia Mathematica Societatis János Bolya, Infinite and Finite Sets
(A. Hajnal, R. Rado, and V. T. Sós, editors), vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 1127–1132.Google Scholar
Nesetril, J. and Rödl, V.,
The Ramsey property for graphs without forbidden complete subgraphs
.
Journal of Combinatorial Theory, (B)
, vol. 20 (1976), pp. 243–249.Google Scholar
Newelski, L.,
Free sets for some special set mappings
.
Israel Journal of Mathematics
, vol. 58 (1987), pp. 213–224.Google Scholar
Newelski, L., Pawlikowski, J., and Seredyński, W.,
Infinite free set for small measure set mappings
.
Proceedings of the American Mathematical Society
, vol. 100 (1987), pp. 335–339.Google Scholar
Nosal, E.,
On partition relation for ordinal numbers
.
Journal of the London Mathematical Society, (2)
, vol. 8 (1974), pp. 306–310.Google Scholar
Nosal, E.,
Partition relations for denumerable ordinals
.
Journal of Combinatorial Theory, (B)
, vol. 27 (1979), pp. 190–197.Google Scholar
Prikry, K.,
On a problem of Gillman and Keisler
.
Annals of Mathematical Logic
, vol. 2 (1970), pp. 179–187.Google Scholar
Prikry, K.,
On a problem of Erdős, Hajnal and Rado
.
Discrete Mathematics
, vol. 2 (1972), pp. 51–59.Google Scholar
Prikry, K.,
On a set-theoretic partition problem
.
Duke Mathematical Journal
, vol. 39 (1972), pp. 77–83.Google Scholar
Prikry, K.,
Kurepa’s hypothesis and a problem of Ulam on families of measures
.
Monatshefte für Mathematik
, vol. 81 (1976), pp. 41–57.Google Scholar
Raghavan, D. and Todorcevic, S.,
Suslin trees, the bounding number, and partition relations
.
Israel Journal of Mathematics
, vol. 225 (2018), pp. 771–796.Google Scholar
Rebholz, J. R.,
Some consequences of the morass and diamond
.
Annals of Mathematical Logic
, vol. 7 (1974), pp. 361–385.Google Scholar
Rinot, A.,
Chain conditions of products
.
Bulletin of the Symbolic Logic
, vol. 20 (2014), pp. 293–314.Google Scholar
Rödl, V.,
On the chromatic number of subgraphs of a graph
.
Proceedings of the American Mathematical Society
, vol. 64 (1977), pp. 370–371.Google Scholar
Schipperus, R.,
Countable partition ordinals
.
Annals of Pure and Applied Logic
, vol. 161 (2010), pp. 1195–1215.Google Scholar
Shelah, S.,
A combinatorial problem: stability and order for models and theories in infinitary languages
.
Pacific Journal of Mathematics
, vol. 41 (1972), pp. 247–261.Google Scholar
Shelah, S.,
Notes on combinatorial set theory
.
Israel Journal of Mathematics
, vol. 14 (1973), pp. 262–277.Google Scholar
Shelah, S.,
Notes on partition calculus
,
Colloquia Mathematica Societatis János Bolyai, Infinite and Finite Sets
(A. Hajnal, R. Rado, and V. T. Sós, editors), vol. 10, János Bolyai Mathematical Society, Keszthely, 1973, pp. 1257–1276.Google Scholar
Shelah, S.,
Colouring without triangles and partition relations
.
Israel Journal of Mathematics
, vol. 20 (1975), pp. 1–12.Google Scholar
Shelah, S.,
A compactness theorem in singular cardinals, free algebras, Whitehead problem, and transversals
.
Israel Journal of Mathematics
, vol. 21 (1975), pp. 319–349.Google Scholar
Shelah, S.,
On successors of singular cardinals, Logic Coll
(M. Boffa, D. van Dalen, and K. McAloon, editors), vol. 78, North-Holland, Amsterdam, 1979, pp. 357–380.Google Scholar
Shelah, S.,
Independence of strong partition relation for small cardinals and the free-subset problem
.
Journal of Symbolic Logic
, vol. 45 (1980), pp. 505–509.Google Scholar
Shelah, S.,
${\mathrm{\aleph}}_{\omega }$
may have a strong partition relation
.
Israel Journal of Mathematics
, vol. 38 (1981), pp. 283–288.Google Scholar
${\mathrm{\aleph}}_{\omega }$
may have a strong partition relation
.
Israel Journal of Mathematics
, vol. 38 (1981), pp. 283–288.Google ScholarShelah, S.,
Proper Forcing
, Lecture Notes in Mathematics, 940, Springer, Heidelberg, London, New York, 1982.Google Scholar
Shelah, S., Iterated forcings and normal ideals on
${\unicode{x3c9}}_1$
.
Israel Journal of Mathematics
, vol. 60 (1987), pp. 345–380.Google Scholar
${\unicode{x3c9}}_1$
.
Israel Journal of Mathematics
, vol. 60 (1987), pp. 345–380.Google ScholarShelah, S.,
Was Sierpiński right?
Israel Journal of Mathematics
, vol. 62 (1988), pp. 355–380.Google Scholar
Shelah, S.,
Consistency of positive partition theorems for graphs and models
,
Set Theory and its Applications
(J. Stepr
$\bar{\mathrm{a}}$
ns, editor), Lecture Notes in Mathematics, 1401, “Set Theory and its Applications”, Proc. Conf. at York University, Toronto, ON, 1987, 1989, pp. 167–193. (sh289)Google Scholar
$\bar{\mathrm{a}}$
ns, editor), Lecture Notes in Mathematics, 1401, “Set Theory and its Applications”, Proc. Conf. at York University, Toronto, ON, 1987, 1989, pp. 167–193. (sh289)Google ScholarShelah, S.,
Incompactness for chromatic numbers of graphs
,
A Tribute to Paul Erdős
,(A. Baker, B. Bollobás, and A. Hajnal, editors), Cambridge University Press, Cambridge, 1990, pp. 361–371.Google Scholar
Shelah, S.,
Strong partition relations below the power set: Consistency, was Sierpi ń ski right, II?
,
Proc Conference on Set Theory and its Applications in honor of A.Hajnal and V.T.Sós
(G. Halász, L. Lovász, D. Miklós, T. Szőnyi, editor), vol. 1/91, János Bolyai Mathematical Society, Budapest, 1991, pp. 637–638.Google Scholar
Shelah, S., On CH+
${2}^{{\mathrm{\aleph}}_1}\to {\left(\unicode{x3b1} \right)}_2^2$
for
$\unicode{x3b1} <{\unicode{x3c9}}_2$
,
Logic Coll. '90
(J. Oikkonen and J. Väänänen, editors), Ass. Symb. Logic, Lect Notes in Logic, 2, pp. 281–289.Google Scholar
${2}^{{\mathrm{\aleph}}_1}\to {\left(\unicode{x3b1} \right)}_2^2$
for
$\unicode{x3b1} <{\unicode{x3c9}}_2$
,
Logic Coll. '90
(J. Oikkonen and J. Väänänen, editors), Ass. Symb. Logic, Lect Notes in Logic, 2, pp. 281–289.Google ScholarShelah, S.,
Proper and Improper Forcing
, Springer, Heidelberg, London, New York, 1998.Google Scholar
Shelah, S.,
Was Sierpi ń ski right? IV
.
Journal of Symbolic Logic
, vol. 65 (2000), pp. 1031–1054.Google Scholar
Shelah, S.,
A partition relation using strongly compact cardinals
.
Proceedings of the American Mathematical Society
, vol. 131 (2003), pp. 2585–2592.Google Scholar
Shelah, S.,
The Erdős-Rado arrow for singular
.
Canadian Mathematical Bulletin
, vol. 52 (2009), pp. 127–131.Google Scholar
Shelah, S.,
Universality among graphs omitting a complete bipartite graph
.
Combinatorica
, vol. 32 (2012), pp. 325–362.Google Scholar
Shelah, S. and Stanley, L.,
A theorem and some consistency results in partition calculus
.
Annals of Pure and Applied Logic
, vol. 36 (1987), pp. 119–152.Google Scholar
Shelah, S. and Stanley, L.,
More consistency results in partition calculus
.
Israel Journal of Mathematics
, vol. 81 (1993), pp. 97–110.Google Scholar
Shore, R.,
Square bracket relations in L
.
Fundamenta Mathematicæ
, vol. 84 (1974), pp. 101–106.Google Scholar
Specker, E.,
Teilmengen von Mengen mit Relationen
.
Commentarii Mathematici Helvetici
, vol. 31 (1957), pp. 302–314.Google Scholar
Stanley, L, Velleman, D., and Morgan, C.,
${\omega}_3{\omega}_1\to {\left({\omega}_3{\omega}_1,3\right)}^2$
requires an inaccessible
.
Proceedings of the American Mathematical Society
, vol. 111 (1991), pp. 1105–1118.Google Scholar
${\omega}_3{\omega}_1\to {\left({\omega}_3{\omega}_1,3\right)}^2$
requires an inaccessible
.
Proceedings of the American Mathematical Society
, vol. 111 (1991), pp. 1105–1118.Google ScholarSteel, J. R. and >van Wesep, R. A.,
Two consequences of determinacy consistent with choice
.
Transactions of the American Mathematical Society
, vol. 272 (1982), pp. 67–85.van+Wesep,+R.+A.,+Two+consequences+of+determinacy+consistent+with+choice+.+Transactions+of+the+American+Mathematical+Society+,+vol.+272+(1982),+pp.+67–85.>Google Scholar
Contests in Higher Mathematics
, (G. J. Székely, editor), Problem Books in Mathematics, Springer, Heidelberg, London, New York, 1996.Google Scholar
Alan, D.,
Taylor: Regularity properties of ideals and ultrafilters
.
Annals of Mathematical Logic
, vol. 16 (1979), pp 33–55.Google Scholar
Alan, D.,
Taylor: On saturated sets of ordinals and Ulam’s problem
.
Fundamenta Mathematicæ
, vol. 109 (1980), pp. 37–53.Google Scholar
Thomassen, C.,
Cycles in graphs of uncountable chromatic number
.
Combinatorica
, vol. 3 (1983), pp. 133–134.Google Scholar
Todorcevic, S.,
Trees, subtrees, and order types
.
Annals of Mathematical Logic
, vol. 20 (1981), pp. 233–268.Google Scholar
Todorcevic, S.,
Reals and positive partition relations
,
Logic Methodology, and Philosophy of Science, VII, Salzburg, 1983
(R. B. Marcus, G. J. W. Dorn, and P. Weingartner, editors), Studies of Logic, 114, North-Holland, Amsterdam, 1986, pp. 159–169.Google Scholar
Todorcevic, S.,
Forcing positive partition relations
.
Transactions of the American Mathematical Society
, vol. 280 (1983), pp. 703–720.Google Scholar
Todorcevic, S.,
On a conjecture of R. Rado
.
Journal of the London Mathematical Society
, vol. 27 (1983), pp. 1–8.Google Scholar
Todorcevic, S.,
Trees and linearly ordered sets
,
Handbook of Set-Theoretic Topology
(K. Kunen and J. Vaughan, editors), North-Holland, Amsterdam, 1984, pp. 235–293.Google Scholar
Todorcevic, S.,
Aronszajn trees and partitions
.
Israel Journal of Mathematics
, vol. 52 (1985), pp. 53–58.Google Scholar
Todorcevic, S.,
Partition relations for partially ordered set
.
Acta Mathematica
, vol. 155 (1985), pp. 1–25.Google Scholar
Todorcevic, S.,
Partitioning pairs of countable ordinals
.
Acta Mathematica
, vol. 159 (1987), pp. 261–294.Google Scholar
Todorcevic, S.,
Partittion problems in topology
,
Contemporary Mathematics
, vol. 84 (1989), pp. 1–116.Google Scholar
Yodorcevic, S.,
Partitioning pairs of countable sets
.
Proceedings of the American Mathematical Society
(S. Todorcevic, editor), vol. 111 (1991), pp. 841–844.Google Scholar
Todorcevic, S.,
Some partitions of three-dimensional combinatorial cubes
.
Journal of Combinatorial Theory, (A)
, vol. 68 (1994), pp. 410–437.Google Scholar
Todorcevic, S.,
Countable chain condition in partition calculus
.
Discrete Mathematics
, vol. 88 (1998), pp. 105–223.Google Scholar
Todorcevic, S.,
Walks om Ordinals and their Characteristics
, Progress in Mathematics, 263, Birkhäuser, 2007.Google Scholar
Todorcevic, S.,
Introduction to Ramsey Spaces
, Annals of Mathematics Studies, Princeton University Press, 2010.Google Scholar
Ulam, St. M.,
A Collection of Mathematical Problems, New York, Interscience Publications, 1960, reprinted
,
Problems in Modern Mathematics
, Dover, Mineola, NY, 2004.Google Scholar
Unger, S., Compactness for the chromatic number at
${\aleph}_{\omega_1+1}$
, manuscript.Google Scholar
${\aleph}_{\omega_1+1}$
, manuscript.Google ScholarHugh Woodin, W.,
The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal
, Walter de Gruyter, 1999.Google Scholar