Homework 1 for
MATH 497A, Introduction to Ramsey Theory
Solutions
Problem 1
Use Ramseys theorem for graphs to show that for every positive integer k there exists
a number N (k) such that if a1 , a2 , . . . , aN (k) is a sequence of N (k) integers, it has
Math 497A Homework 9 Fall 2008 Due: Friday, November 7
(1) Let E be an elliptic curve dened over Fq (with q = pr ), and assume that E[p] = Z/pZ. Show that the Frobenius endomorphism q is not multiplication-by-m for any integer m. (2) Let E be an elliptic
Homework 1 for
MATH 497A, Introduction to Ramsey Theory
Solutions
Problem 1
Use Ramseys theorem for graphs to show that for every positive integer k there exists
a number N (k) such that if a1 , a2 , . . . , aN (k) is a sequence of N (k) integers, it has
Homework 2 for
MATH 497A, Introduction to Ramsey Theory
Due: Wednesday September 7
Problem 1
Show that the Ramsey numbers R(m, n) (really R(m, n; 2) in light of Problem 2) satisfy the
bound
m+n2
R(m, n)
m1
for all m, n 1. (Hint: Exploit the familiar recu
Homework 3 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 12
Problem 1
A geometric application of Turns Theorem.
Let S
2
with d the usual Euclidean distance. The diameter of S is given by
d (S ) = supcfw_d ( x , y ) : x , y S .
Assume
Homework 4 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 19
Problem 1
Upper Bounds for Ramseys Theorem
From the various proofs of Ramseys Theorem, try to extract an upper bound (as sharp
as you can) on R( p; k; r ). Recall that R( p;
Homework 5 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 26
Problem 1 The Axiom of Regularity
The Axiom of Regularity is formalized as
If S = , then there exists an x S , such that x S = .
Show that if one assumes the Axiom of Regular
Homework 6 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday October 3
Problem 1
Failure of Ramseys Theorem for innite colorings
Show that for any innite cardinal , 2 (3)2 .
Solution. This is in the lecture notes from 09/26.
Problem 2
Failure of Ra
Homework 10 for
MATH 497A, Introduction to Ramsey Theory
Solutions
Problem 1 Non-standard models of arithmetic, part I
Consider the language L = cfw_S , +, 0, where S is a unary function symbol, + is a binary function symbol, and
0 is a constant symbol.
C
Midterm Preparation for
MATH 497A, Introduction to Ramsey Theory
Denitions and Theorems
Review the denitions and theorems covered so far. You will be asked to state a few of them
precisely.
Theorem from Class
You will be asked to state and prove one or tw
Homework 10 for
MATH 497A, Introduction to Ramsey Theory
Solutions
Problem 1 Non-standard models of arithmetic, part I
Consider the language L = cfw_S , +, 0, where S is a unary function symbol, + is a binary function symbol, and
0 is a constant symbol.
C
Homework 6 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday October 3
Problem 1
Failure of Ramseys Theorem for innite colorings
Show that for any innite cardinal , 2 (3)2 .
Solution. This is in the lecture notes from 09/26.
Problem 2
Failure of Ra
Homework 5 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 26
Problem 1 The Axiom of Regularity
The Axiom of Regularity is formalized as
If S = , then there exists an x S , such that x S = .
Show that if one assumes the Axiom of Regular
Homework 4 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 19
Problem 1
Upper Bounds for Ramseys Theorem
From the various proofs of Ramseys Theorem, try to extract an upper bound (as sharp
as you can) on R( p; k; r ). Recall that R( p;
Homework 3 for
MATH 497A, Introduction to Ramsey Theory
Due: Monday September 12
Problem 1
A geometric application of Turns Theorem.
Let S
2
with d the usual Euclidean distance. The diameter of S is given by
d (S ) = supcfw_d ( x , y ) : x , y S .
Assume
Homework 2 for
MATH 497A, Introduction to Ramsey Theory
Due: Wednesday September 7
Problem 1
Show that the Ramsey numbers R(m, n) (really R(m, n; 2) in light of Problem 2) satisfy the
bound
m+n2
R(m, n)
m1
for all m, n 1. (Hint: Exploit the familiar recu
Math 497A Homework 2 Fall 2008 Due: Friday, September 12
(1) Let d be a nonzero rational number. We have seen in class that the curve E : X 3 + Y 3 = dZ 3 is an elliptic curve over Q. We dene the group law on E so that the point O = [1, 1, 0] becomes the