Ramsey%20Theory%20on%20the%20Integers%20and%20Reals

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Ramsey Theory on the Integers and Reals

• Daniel J. Kleitman and Jacob Fox
• MIT

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Schurs Theorem (1916)

• In every coloring of the positive integers with
  finitely
• many colors, there exists x, y, and z all the
  same
• color such that x y z.
• The following 3-coloring of the integers 1,13
  does
• not have a monochromatic solution to x y z
• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
• However, every 3-coloring of the integers 1,14
  has a
• monochromatic solution to x y z.

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Partition Regularity

• A linear homogeneous equation
• a1x1 a2x2 a3x3 0 (1)
• with integer coefficients is called r-regular if
  every
• r-coloring of the positive integers has a
• monochromatic solution to Equation (1).
• Equation (1) is called regular if it is r-regular
  for all
• positive integers r.
• Example Schurs theorem implies the equation
• xyz is regular.

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The Equation x1 2x2 5x3 0

• Every 3-coloring of the integers 1,45 has a
• monochromatic solution to x1 2x2 5x3 0.
• Therefore, the equation x1 2x2 5x3 0 is
• 3-regular.

Richard K. Guy, Unsolved problems in number
theory. Third edition. Problem Books in
Mathematics. Springer-Verlag, New York, 2004.
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The Equation x1 2x2 5x3 0(Continued)

• If we color each positive integer n m5k where 5
  is
• not a factor of m by the remainder when m is
• divided by 5, then there are no monochromatic
• solutions to x1 2x2 5x3 0 in this
  4-coloring of
• the positive integers.
• Therefore, the equation x1 2x2 5x3 0 is
• 3-regular, but not 4-regular.

Richard K. Guy, Unsolved problems in number
theory. Third edition. Problem Books in
Mathematics. Springer-Verlag, New York, 2004.
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Rados Theorem (1933)

• Richard Rados thesis Studien zur Kombinatorik
• generalized Schurs theorem by classifying those
• finite linear equations that are regular.

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Studien zur Kombinatorik (1933)
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Rados Theorem (1933)

• The equation a1x1a2x2 anxn 0 is
• regular if and only if some subset of the
• non-zero coefficients sums to 0.

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Rados Boundedness Conjecture (1933)

• For every positive integer n, there exists an
• integer kk(n) such that every linear
• homogeneous equation a1x1a2x2 anxn0
• that is k-regular is regular.
• Rado proved his conjecture in the trivial cases
• n 1 and n 2. Until recently, the conjecture
• has been open for n gt 2.

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Fox-Kleitman Theorem

• Every 24-regular linear homogeneous equation
  
• a1 ax2 ax3 0 is regular.

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Partition Regularity over R

• A linear homogeneous equation
• a1x1 a2x2 a3x3 0 (1)
• with real coefficients is called r-regular over R
• if every r-coloring of the nonzero real numbers
• has a monochromatic solution to Equation (1).
• A linear homogeneous equation is called
• regular over R if it is r-regular over R for all
• positive integers r.

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Rados Theorem over R (1943)

• The equation a1x1a2x2 anxn 0 is
• regular over R if and only if some subset of
• the non-zero coefficients sums to 0.
• Regular examples
• x1 ?x2 - (1 ? )x3 0
• ? x1 - ? x2 4x3 0
• Nonregular example x1 2x2 - 4x3 0

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The equation x1 2x2 - 4x3 0

• Let T denote the statement
• the equation x1 2x2 - 4x3 0 is 3-regular
  over R.
• Jacob Fox and Rados Radoicic recently proved the
• statement T is independent of the
  Zermelo-Fraenkel
• axioms for set theory.
• That is, no contradiction arises if you assume T
  is true
• and no contradiction arises if you assume T is
  false.

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Detour Infinite numbers (Cardinals)

• We now assume the axiom of choice
• for every family C of nonempty sets, there exists
  a
• function f defined on C such that f(S) is an
  element
• of S for every S from C.
• Two sets A and B are said to have the same size
• if there exists a bijective function f A ? B.
  The
• cardinality of a set S is the size of S.
• The cardinality of a,b,c,d is 4.
• The cardinality of N is denoted by ?0.
• The cardinality of R is denoted by c.

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The cardinals

• The cardinal numbers (in increasing order)
• 0, 1, 2, , ?0, ?1, ?2, , ??, ??1,
• In 1873, Cantor proved that c gt ?0.
• So which one of the cardinals is c?

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What is the Cardinality of the Continuum?

• Are there any cardinals between ?0 and c?
• In other words, does c ?1? This is known as
• the continuum hypothesis. Cantor spent ten
• years of his life unsuccessfully trying to prove
• the continuum hypothesis. It is believed that
• this contributed to his mental illness later in
  life.

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The Cardinality of the Continuum

• In 1937, Kurt Gödel proved that the continuum
• hypothesis can not be proved false.
• In 1963, Paul Cohen proved that the continuum
• hypothesis can not be proved true.
• In fact, for every positive integer n, it is
• independent of ZFC (Zermelo-Fraenkel axioms for
• set theory Axiom of Choice) that c ?n.

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?Countable Regularity

• A linear homogeneous equation
• a1x1 a2x2 a3x3 0 (1)
• with real coefficients is called ?0-regular
• if every coloring of the real numbers by
• positive integers has a monochromatic
• solution to Equation (1) in distinct xi.

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Countable Regularity

• Paul Erdos and Shizuo Kakutani in 1943 proved
• that the negation of the continuum hypothesis is
• equivalent to the equation x1 x2 - x3 - x4 0
  being
• ?0-regular.
• Fox recently classified which linear homogeneous
• equations are ?0-regular in terms of the
  cardinality
• of the continuum.
• For example, c ? ?4 is equivalent to the equation
  
• x1 3x2 - x3 - x4 x5 x6 0 being
  ?0-regular.

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Thank You