Ramsey Theory on the
Integers and Reals
Daniel J. Kleitman and Jacob Fox
MIT
Schur’s 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.
Partition Regularity
A linear homogeneous equation
a 1x 1 + a 2x 2 + … + a 3x 3 = 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: Schur’s theorem implies the equation
x+y=z is regular.
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.
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.
Rado’s Theorem (1933)
Richard Rado’s thesis Studien zur Kombinatorik
generalized Schur’s theorem by classifying those
finite linear equations that are regular.
Studien zur Kombinatorik (1933)
Rado’s Theorem (1933)
The equation a1x1+a2x2+…+ anxn =0 is
regular if and only if some subset of the
non-zero coefficients sums to 0.
Rado’s Boundedness Conjecture (1933)
For every positive integer n, there exists an
integer k=k(n) such that every linear
homogeneous equation a1x1+a2x2+…+ anxn=0
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 > 2.
Fox-Kleitman Theorem
Every 24-regular linear homogeneous equation
a1 + ax2 + ax3 = 0 is regular.
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.
Rado’s Theorem over R (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 = 0
Nonregular example: x1 + 2x2 - 4x3 = 0
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.
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.
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?
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.
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.
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.
Countable Regularity
Paul Erdős 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.