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Partition regularity

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In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.

Given a set , a collection of subsets is called partition regular if every set A in the collection has the property that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any , and any finite partition , there exists an i ≤ n such that belongs to . Ramsey theory is sometimes characterized as the study of which collections are partition regular.

Examples

  • The collection of all infinite subsets of an infinite set X is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.)
  • Sets with positive upper density in : the upper density of is defined as (Szemerédi's theorem)
  • For any ultrafilter on a set , is partition regular: for any , if , then exactly one .
  • Sets of recurrence: a set R of integers is called a set of recurrence if for any measure-preserving transformation of the probability space (Ω, β, μ) and of positive measure there is a nonzero so that .
  • Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then the collection of a.p.-rich subsets is partition regular (Van der Waerden, 1927).
  • Let be the set of all n-subsets of . Let . For each n, is partition regular. (Ramsey, 1930).
  • For each infinite cardinal , the collection of stationary sets of is partition regular. More is true: if is stationary and for some , then some is stationary.
  • The collection of -sets: is a -set if contains the set of differences for some sequence .
  • The set of barriers on : call a collection of finite subsets of a barrier if:
    • and
    • for all infinite , there is some such that the elements of X are the smallest elements of I; i.e. and .
This generalizes Ramsey's theorem, as each is a barrier. (Nash-Williams, 1965)[1]

Diophantine equations

A Diophantine equation is called partition regular if the collection of all infinite subsets of containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations are partition regular. Much progress has been made recently on classifying nonlinear Diophantine equations.[7][8]

References

  1. ^ Nash-Williams, C. St. J. A. (1965). "On well-quasi-ordering transfinite sequences". Mathematical Proceedings of the Cambridge Philosophical Society. 61 (1): 33–39. Bibcode:1965PCPS...61...33N. doi:10.1017/S0305004100038603.
  2. ^ Brown, Thomas Craig (1971). "An interesting combinatorial method in the theory of locally finite semigroups". Pacific Journal of Mathematics. 36 (2): 285–289. doi:10.2140/pjm.1971.36.285.
  3. ^ Sanders, Jon Henry (1968). A Generalization of Schur's Theorem, Doctoral Dissertation (PhD). Yale University.
  4. ^ Deuber, Walter (1973). "Partitionen und lineare Gleichungssysteme". Mathematische Zeitschrift. 133 (2): 109–123. doi:10.1007/BF01237897.
  5. ^ Hindman, Neil (1974). "Finite sums from sequences within cells of a partition of ". Journal of Combinatorial Theory. Series A. 17 (1): 1–11. doi:10.1016/0097-3165(74)90023-5.
  6. ^ Hindman, Neil; Strauss, Dona (1998). Algebra in the Stone–Čech compactification. De Gruyter. doi:10.1515/9783110258356. ISBN 978-3-11-025623-9.
  7. ^ Di Nasso, Mauro; Luperi Baglini, Lorenzo (January 2018). "Ramsey properties of nonlinear Diophantine equations". Advances in Mathematics. 324: 84–117. arXiv:1606.02056. doi:10.1016/j.aim.2017.11.003. ISSN 0001-8708.
  8. ^ Barrett, Jordan Mitchell; Lupini, Martino; Moreira, Joel (May 2021). "On Rado conditions for nonlinear Diophantine equations". European Journal of Combinatorics. 94 103277. arXiv:1907.06163. doi:10.1016/j.ejc.2020.103277. ISSN 0195-6698.

Further reading