Ramsey Theory Presentation

8/3/2019 Ramsey Theory Presentation

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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 finitelymany colors, there exists x, y, and z all the samecolor such that x + y = z.

The following 3-coloring of the integers [1,13] doesnot 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 amonochromatic 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-regularif everyr-coloring of the positive integers has amonochromatic solution to Equation (1).Equation (1) is called regularif it is r-regular for all

positive integers r.

Example: Schurs theorem implies the equation

x+y=z 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 5x

3=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 a1x1+a2x2++ 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 k=k(n) such that every linear

homogeneous equation a1x1+a2x2++ anxn=0that is k-regular is regular.

Rado proved his conjecture in the trivial casesn = 1 and n = 2. Until recently, the conjecture

has been open for n > 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


a1x1 + a2x2 + + a3x3 = 0 (1)

with real coefficients is called r-regular over Rif every r-coloring of the nonzero real numbers

has a monochromatic solution to Equation (1).

A linear homogeneous equation is calledregular over Rif it is r-regular over R for all

positive integers r.


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Rados Theorem overR (1943)

The equation a1x1+a2x2++ 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 = 0Nonregular example: x1 + 2x2 - 4x3 = 0


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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. Thecardinality 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 > 0.

So which one of the cardinals is c?


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The Cardinality of the Continuum

In 1937, Kurt Gdel 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


a1x1 + a2x2 + + a3x3 = 0 (1)

with real coefficients is called

0-regularif 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 Erds and Shizuo Kakutani in 1943 provedthat the negation of the continuum hypothesis isequivalent to the equation x1 + x2 - x3 - x4 = 0 being

0-regular.

Fox recently classified which linear homogeneousequations are 0-regular in terms of the cardinality

of the continuum.

For example, c4 is equivalent to the equationx1 + 3x2 - x3 - x4 x5 x6 = 0 being 0-regular.


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Ramsey Theory Presentation

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