nonstandard methods in combinatorial number theory · nonstandard methods in combinatorial number...
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Nonstandard methods in combinatorial
number theory
Lorenzo Luperi Baglini
Joint work with Mauro Di Nasso
Wolfgang Pauli Institute
AILA 2017
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 1 / 14
![Page 2: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/2.jpg)
Questions
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y, z P Ai such that x2 � y2 � z2?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that x� y, x � y P Ai?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that xy P Ai?
Is it true that for every A � N if BDpAq ¡ 0 then there exist
in�nite sets X,Y such that X � Y � A?
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 2 / 14
![Page 3: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/3.jpg)
Questions
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y, z P Ai such that x2 � y2 � z2?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that x� y, x � y P Ai?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that xy P Ai?
Is it true that for every A � N if BDpAq ¡ 0 then there exist
in�nite sets X,Y such that X � Y � A?
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 2 / 14
![Page 4: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/4.jpg)
Questions
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y, z P Ai such that x2 � y2 � z2?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that x� y, x � y P Ai?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that xy P Ai?
Is it true that for every A � N if BDpAq ¡ 0 then there exist
in�nite sets X,Y such that X � Y � A?
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 2 / 14
![Page 5: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/5.jpg)
Questions
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y, z P Ai such that x2 � y2 � z2?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that x� y, x � y P Ai?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that xy P Ai?
Is it true that for every A � N if BDpAq ¡ 0 then there exist
in�nite sets X,Y such that X � Y � A?
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 2 / 14
![Page 6: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/6.jpg)
Questions
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y, z P Ai such that x2 � y2 � z2?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that x� y, x � y P Ai?
Is it true that for every �nite partition N � A1 Y � � � YAk there
exist i ¤ k and x, y P Ai such that xy P Ai?
Is it true that for every A � N if BDpAq ¡ 0 then there exist
in�nite sets X,Y such that X � Y � A?
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 2 / 14
![Page 7: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/7.jpg)
Terminology
We say that a polynomial P px1, ..., xnq is partition regular on
N � t1, 2, ...u if whenever the natural numbers are �nitely colored there
exists a monochromatic solution to the equation P px1, ..., xnq � 0.
Theorem (Rado)
Let P px1, ..., xnq �°ni�1 aixi be a linear polynomial. The following
conditions are equivalent:
1 P px1, ..., xnq is partition regular on N;2 there is a nonempty subset J of t1, ..., nu such that
°jPJ
aj � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 3 / 14
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Terminology
We say that a polynomial P px1, ..., xnq is partition regular on
N � t1, 2, ...u if whenever the natural numbers are �nitely colored there
exists a monochromatic solution to the equation P px1, ..., xnq � 0.
Theorem (Rado)
Let P px1, ..., xnq �°ni�1 aixi be a linear polynomial. The following
conditions are equivalent:
1 P px1, ..., xnq is partition regular on N;2 there is a nonempty subset J of t1, ..., nu such that
°jPJ
aj � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 3 / 14
![Page 9: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/9.jpg)
Terminology
We say that a polynomial P px1, ..., xnq is partition regular on
N � t1, 2, ...u if whenever the natural numbers are �nitely colored there
exists a monochromatic solution to the equation P px1, ..., xnq � 0.
Theorem (Rado)
Let P px1, ..., xnq �°ni�1 aixi be a linear polynomial. The following
conditions are equivalent:
1 P px1, ..., xnq is partition regular on N;
2 there is a nonempty subset J of t1, ..., nu such that°jPJ
aj � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 3 / 14
![Page 10: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/10.jpg)
Terminology
We say that a polynomial P px1, ..., xnq is partition regular on
N � t1, 2, ...u if whenever the natural numbers are �nitely colored there
exists a monochromatic solution to the equation P px1, ..., xnq � 0.
Theorem (Rado)
Let P px1, ..., xnq �°ni�1 aixi be a linear polynomial. The following
conditions are equivalent:
1 P px1, ..., xnq is partition regular on N;2 there is a nonempty subset J of t1, ..., nu such that
°jPJ
aj � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 3 / 14
![Page 11: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/11.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 12: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/12.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 13: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/13.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;
2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 14: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/14.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;
3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 15: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/15.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;
4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 16: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/16.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 17: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/17.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 18: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/18.jpg)
Ultra�lters on N
De�nition
A family U � ℘pNq is an ultra�lter if the following conditions hold:
1 H R U ;2 A P U , A � B ñ B P U ;3 A,B P U ñ AXB P U ;4 for every A � N A P U or Ac P U .
De�nition
A P U ` V ô tn P N | A� n P Vu P U .
Fact: Since the '70s, ultra�lters have been one of the main tools to
prove results in combinatorial number theory.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 4 / 14
![Page 19: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/19.jpg)
Translation in terms of Ultra�lters
De�nition
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on N. We say
that U is a σP-ultra�lter if and only if for every set A P U there are
a1, ..., an P A such that P pa1, .., anq � 0.
Theorem
P px1, ..., xnq is partition regular if and only if there exists a
σP -ultra�lter.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 5 / 14
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Translation in terms of Ultra�lters
De�nition
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on N. We say
that U is a σP-ultra�lter if and only if for every set A P U there are
a1, ..., an P A such that P pa1, .., anq � 0.
Theorem
P px1, ..., xnq is partition regular if and only if there exists a
σP -ultra�lter.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 5 / 14
![Page 21: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/21.jpg)
Translation in terms of Ultra�lters
De�nition
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on N. We say
that U is a σP-ultra�lter if and only if for every set A P U there are
a1, ..., an P A such that P pa1, .., anq � 0.
Theorem
P px1, ..., xnq is partition regular if and only if there exists a
σP -ultra�lter.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 5 / 14
![Page 22: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/22.jpg)
Sets of Generators of U
Let �N be an hyperextension of N satisfying the c�-enlarging property.
De�nition
Given an ultra�lter U on N, its set of generators is
GU � tα P�N | U � Uαu,
where Uα � tA � N | α P�Au.
If we work in an extension that allows for the iteration of the �-map, we
also have that
Uα ` Uβ � U�β�α,Uα d Uβ � U�β�α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 6 / 14
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Sets of Generators of U
Let �N be an hyperextension of N satisfying the c�-enlarging property.
De�nition
Given an ultra�lter U on N, its set of generators is
GU � tα P�N | U � Uαu,
where Uα � tA � N | α P�Au.
If we work in an extension that allows for the iteration of the �-map, we
also have that
Uα ` Uβ � U�β�α,Uα d Uβ � U�β�α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 6 / 14
![Page 24: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/24.jpg)
Sets of Generators of U
Let �N be an hyperextension of N satisfying the c�-enlarging property.
De�nition
Given an ultra�lter U on N, its set of generators is
GU � tα P�N | U � Uαu,
where Uα � tA � N | α P�Au.
If we work in an extension that allows for the iteration of the �-map, we
also have that
Uα ` Uβ � U�β�α,Uα d Uβ � U�β�α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 6 / 14
![Page 25: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/25.jpg)
The translation in nonstandard terms
Theorem
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on βN. The
following two conditions are equivalent:
1 U is a σP -ultra�lter;
2 there are elements α1, ..., αn in GU such that P pα1, ..., αnq � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 7 / 14
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The translation in nonstandard terms
Theorem
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on βN. The
following two conditions are equivalent:
1 U is a σP -ultra�lter;
2 there are elements α1, ..., αn in GU such that P pα1, ..., αnq � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 7 / 14
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The translation in nonstandard terms
Theorem
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on βN. The
following two conditions are equivalent:
1 U is a σP -ultra�lter;
2 there are elements α1, ..., αn in GU such that P pα1, ..., αnq � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 7 / 14
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The translation in nonstandard terms
Theorem
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on βN. The
following two conditions are equivalent:
1 U is a σP -ultra�lter;
2 there are elements α1, ..., αn in GU such that P pα1, ..., αnq � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 7 / 14
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The translation in nonstandard terms
Theorem
Let P px1, ..., xnq be a polynomial, and U an ultra�lter on βN. The
following two conditions are equivalent:
1 U is a σP -ultra�lter;
2 there are elements α1, ..., αn in GU such that P pα1, ..., αnq � 0.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 7 / 14
![Page 30: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/30.jpg)
Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U .
Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,
and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter.
Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ.
Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Examples
Theorem (Schur)
The polynomial P px, y, zq : x� y � z is partition regular.
Proof: Let U be an additive idempotent ultra�lter, and α P�N a
generator of U . Then �α P GU and α��α P GU ,and
P pα, �α, α� �αq � 0.
Theorem (Hindman)
The polynomial P px, y, u, vq : x� y � uv is partition regular.
Proof: Let U be a multiplicatively idempotent Schur's ultra�lter. Let
α, β, γ P GU be such that α� β � γ. Then
α � �α� β � �α � γ � �α
and we conclude setting x � α � �α, y � β � �α, u � α, v � �α.Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 8 / 14
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Main Partition Regularity Result/1
TheoremLet F be the family of functions whose PR on N is witnessed by at least an ultra�lter
U P Ipdq XKpdq X Ip`q X BD.
Then F includes:
Every Rado polynomial
c1x1 � . . .� cnxn � P py1, . . . , ykq
with injectivity |tx1, . . . , xnu| ¥ n� 1 and |ty1, . . . , yku| � k, and with
injectivity |tx1, x2u| � 2 when n � 2 and k � 1, and with full injectivity
|tx1, . . . , xn, y1, . . . , yku| � n� k when P � 0 is linear ;
Every polynomial of the form
n
i�1
ci xi
�¹jPFi
yj
�
where°n
i�1 ci xi is a Rado polynomial and sets Fi � t1, . . . ,mu, with full
injectivity when n ¡ 2, and with injectivity |tx1, x2, y1, . . . , ymu| ¥ m� 1 when
n � 2 ;
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 9 / 14
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Main Partition Regularity Result/1
TheoremLet F be the family of functions whose PR on N is witnessed by at least an ultra�lter
U P Ipdq XKpdq X Ip`q X BD.
Then F includes:
Every Rado polynomial
c1x1 � . . .� cnxn � P py1, . . . , ykq
with injectivity |tx1, . . . , xnu| ¥ n� 1 and |ty1, . . . , yku| � k, and with
injectivity |tx1, x2u| � 2 when n � 2 and k � 1, and with full injectivity
|tx1, . . . , xn, y1, . . . , yku| � n� k when P � 0 is linear ;
Every polynomial of the form
n
i�1
ci xi
�¹jPFi
yj
�
where°n
i�1 ci xi is a Rado polynomial and sets Fi � t1, . . . ,mu, with full
injectivity when n ¡ 2, and with injectivity |tx1, x2, y1, . . . , ymu| ¥ m� 1 when
n � 2 ;
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 9 / 14
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Main Partition Regularity Result/1
TheoremLet F be the family of functions whose PR on N is witnessed by at least an ultra�lter
U P Ipdq XKpdq X Ip`q X BD.
Then F includes:
Every Rado polynomial
c1x1 � . . .� cnxn � P py1, . . . , ykq
with injectivity |tx1, . . . , xnu| ¥ n� 1 and |ty1, . . . , yku| � k, and with
injectivity |tx1, x2u| � 2 when n � 2 and k � 1, and with full injectivity
|tx1, . . . , xn, y1, . . . , yku| � n� k when P � 0 is linear ;
Every polynomial of the form
n
i�1
ci xi
�¹jPFi
yj
�
where°n
i�1 ci xi is a Rado polynomial and sets Fi � t1, . . . ,mu, with full
injectivity when n ¡ 2, and with injectivity |tx1, x2, y1, . . . , ymu| ¥ m� 1 when
n � 2 ;
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 9 / 14
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Main Partition Regularity Result/1
TheoremLet F be the family of functions whose PR on N is witnessed by at least an ultra�lter
U P Ipdq XKpdq X Ip`q X BD.
Then F includes:
Every Rado polynomial
c1x1 � . . .� cnxn � P py1, . . . , ykq
with injectivity |tx1, . . . , xnu| ¥ n� 1 and |ty1, . . . , yku| � k, and with
injectivity |tx1, x2u| � 2 when n � 2 and k � 1, and with full injectivity
|tx1, . . . , xn, y1, . . . , yku| � n� k when P � 0 is linear ;
Every polynomial of the form
n
i�1
ci xi
�¹jPFi
yj
�
where°n
i�1 ci xi is a Rado polynomial and sets Fi � t1, . . . ,mu, with full
injectivity when n ¡ 2, and with injectivity |tx1, x2, y1, . . . , ymu| ¥ m� 1 when
n � 2 ;
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 9 / 14
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Main Partition Regularity Result/2
Theorem
Every function f of the form
fpx, y1, . . . , ykq � x�k¹i�1
yi
with full injectivity |tx, y1, . . . , yku| � k � 1;
Every function f of the form
fpx, y1, . . . , ykq � x�k¹i�1
yaii
with full injectivity |tx, y1, . . . , yku| � k � 1, whenever the
exponents ai P Z satisfy°ni�1 ai � 1.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 10 / 14
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Main Partition Regularity Result/2
Theorem
Every function f of the form
fpx, y1, . . . , ykq � x�k¹i�1
yi
with full injectivity |tx, y1, . . . , yku| � k � 1;
Every function f of the form
fpx, y1, . . . , ykq � x�k¹i�1
yaii
with full injectivity |tx, y1, . . . , yku| � k � 1, whenever the
exponents ai P Z satisfy°ni�1 ai � 1.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 10 / 14
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Main Partition Regularity Result/3
Moreover, the family F satis�es a technical closure properties.
Example: U witnesses the partition regularity of
$'&'%
x1 � x2 � 7x3
t � y21y�12
x1 � t
So we get the partition regularity of the equation
x2 � 7x3 � y21y�12 .
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 11 / 14
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Main Partition Regularity Result/3
Moreover, the family F satis�es a technical closure properties.
Example: U witnesses the partition regularity of
$'&'%
x1 � x2 � 7x3
t � y21y�12
x1 � t
So we get the partition regularity of the equation
x2 � 7x3 � y21y�12 .
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 11 / 14
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Main Partition Regularity Result/3
Moreover, the family F satis�es a technical closure properties.
Example: U witnesses the partition regularity of
$'&'%
x1 � x2 � 7x3
t � y21y�12
x1 � t
So we get the partition regularity of the equation
x2 � 7x3 � y21y�12 .
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 11 / 14
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Main Partition Regularity Result/3
Moreover, the family F satis�es a technical closure properties.
Example: U witnesses the partition regularity of
$'&'%
x1 � x2 � 7x3
t � y21y�12
x1 � t
So we get the partition regularity of the equation
x2 � 7x3 � y21y�12 .
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 11 / 14
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Main Partition Regularity Result/3
Moreover, the family F satis�es a technical closure properties.
Example: U witnesses the partition regularity of
$'&'%
x1 � x2 � 7x3
t � y21y�12
x1 � t
So we get the partition regularity of the equation
x2 � 7x3 � y21y�12 .
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 11 / 14
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Non-partition regularity/1
We use the p-expansions of numbers and many tools of nonstandard
analysis.
Theorem
Let P pxq �°α cαx
α P Zrx1, . . . , xns be an homogeneous polynomial.If
for every nonempty J � supppP q one has°αPJ cα � 0, then P pxq is
not PR.
Example: x2 � y2 � 3z2 is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 12 / 14
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Non-partition regularity/1
We use the p-expansions of numbers and many tools of nonstandard
analysis.
Theorem
Let P pxq �°α cαx
α P Zrx1, . . . , xns be an homogeneous polynomial.
If
for every nonempty J � supppP q one has°αPJ cα � 0, then P pxq is
not PR.
Example: x2 � y2 � 3z2 is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 12 / 14
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Non-partition regularity/1
We use the p-expansions of numbers and many tools of nonstandard
analysis.
Theorem
Let P pxq �°α cαx
α P Zrx1, . . . , xns be an homogeneous polynomial.If
for every nonempty J � supppP q one has°αPJ cα � 0, then P pxq is
not PR.
Example: x2 � y2 � 3z2 is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 12 / 14
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Non-partition regularity/1
We use the p-expansions of numbers and many tools of nonstandard
analysis.
Theorem
Let P pxq �°α cαx
α P Zrx1, . . . , xns be an homogeneous polynomial.If
for every nonempty J � supppP q one has°αPJ cα � 0, then P pxq is
not PR.
Example: x2 � y2 � 3z2 is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 12 / 14
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Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term.
If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
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Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term. If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
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Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term. If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
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Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term. If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
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Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term. If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
![Page 60: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/60.jpg)
Non partition regularity/2
Theorem
For every i � 1, . . . , n let Pipxiq �°dis�1 ci,sx
si be a polynomial of degree
di in the variable xi with no constant term. If the Diophantine equation
n
i�1
Pipxiq � 0
is PR then the following �Rado's condition� is satis�ed:
�There exists a nonempty set J � t1, . . . , nu such that di � dj for
every i, j P J , and°jPJ cj,dj � 0.�
Example:
1 The polynomial P px, yq � x3 � 2x� y3 � 2y is not PR.
2 If k R tn,mu, the polynomial xn � ym � zk is not PR.
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 13 / 14
![Page 61: Nonstandard methods in combinatorial number theory · Nonstandard methods in combinatorial number theory Lorenzo Luperi Baglini Joint work with Mauro Di Nasso Wolfgang Pauli Institute](https://reader034.vdocuments.site/reader034/viewer/2022042403/5f1570ca1a87fd15ef52cbe1/html5/thumbnails/61.jpg)
Thank You!
email: [email protected]
Lorenzo Luperi Baglini Wolfgang Pauli Institute 27 September 2017 14 / 14