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.

ExamplesEdit

  • 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)
  • finite products of infinite trees (Halpern–Läuchli, 1966)
  • piecewise syndetic sets (Brown, 1968)
  • Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finite sums. Then the collection of i.p.-rich subsets is partition regular (FolkmanRado–Sanders, 1968).
  • (m, p, c)-sets (Deuber, 1973)
  • IP sets (Hindman, 1974, see also Hindman, Strauss, 1998)
  • MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)
  • central sets; i.e. the members of any minimal idempotent in  , the Stone–Čech compactification of the integers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)

Diophantine equationsEdit

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.[1][2]

ReferencesEdit

  1. ^ 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.
  2. ^ 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.
  1. Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory A 93 (2001), 18–36.
  2. T. Brown, An interesting combinatorial method in the theory of locally finite semigroups, Pacific J. Math. 36, no. 2 (1971), 285–289.
  3. W. Deuber, Mathematische Zeitschrift 133, (1973) 109–123
  4. N. Hindman, Finite sums from sequences within cells of a partition of N, J. Comb. Theory A 17 (1974) 1–11.
  5. C.St.J.A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–39.
  6. N. Hindman, D. Strauss, Algebra in the Stone–Čech compactification, De Gruyter, 1998
  7. J.Sanders, A Generalization of Schur's Theorem, Doctoral Dissertation, Yale University, 1968.