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[edit]

  • 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
  • For any ultrafilter on a set , is partition regular. If , then for exactly one is .
  • 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)

References[edit]

  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.