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2nd lecture, combinatorics, bits pilani
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Combinatorial Mathematics
T. Singh
Department of MathematicsBITS Pilani KK Birla Goa, Campus Goa
5th August 2015
Combinatorial Principle
Combinatorial principle that can be used to solve a variety ofinteresting problems. This principle is known as PHP, Dirichletdrawer principle and shoebox principle.
Theorem
PHP in simple form If n + 1 objects are put into n boxes, then atleast one box contains two are more of the objects.
Remark
Note that neither the PHP nor its proof gives any help in findingthe box that contains two or more objects.
Whenever PHP is applied to prove the existence of an arrangementor some phenomenon, it will give no indication of how to constructthe arrangement.
T. Singh Introduction
Combinatorial Principle
Combinatorial principle that can be used to solve a variety ofinteresting problems. This principle is known as PHP, Dirichletdrawer principle and shoebox principle.
Theorem
PHP in simple form If n + 1 objects are put into n boxes, then atleast one box contains two are more of the objects.
Remark
Note that neither the PHP nor its proof gives any help in findingthe box that contains two or more objects.
Whenever PHP is applied to prove the existence of an arrangementor some phenomenon, it will give no indication of how to constructthe arrangement.
T. Singh Introduction
Combinatorial Principle
Combinatorial principle that can be used to solve a variety ofinteresting problems. This principle is known as PHP, Dirichletdrawer principle and shoebox principle.
Theorem
PHP in simple form If n + 1 objects are put into n boxes, then atleast one box contains two are more of the objects.
Remark
Note that neither the PHP nor its proof gives any help in findingthe box that contains two or more objects.
Whenever PHP is applied to prove the existence of an arrangementor some phenomenon, it will give no indication of how to constructthe arrangement.
T. Singh Introduction
PHP contd.
There are other principles related to the PHP that are worthstating:
If n objects are put into n boxes and no box is empty, theneach box contains exactly one object.
If n objects are put into n box and no box gets more than oneobject, then each box has an object in it.
T. Singh Introduction
PHP contd.
There are other principles related to the PHP that are worthstating:
If n objects are put into n boxes and no box is empty, theneach box contains exactly one object.
If n objects are put into n box and no box gets more than oneobject, then each box has an object in it.
T. Singh Introduction
More abstract formulations of the three principles enunciated thusfar are the following.Let f : X → Y be a function from the set X to Y .
If X has more elements that Y then f is not one to one.
If X and Y have the same number of elements and f is ontothen f is one to one.
If X and Y have the same number of elements and f is one toone then f is onto.
T. Singh Introduction
More abstract formulations of the three principles enunciated thusfar are the following.Let f : X → Y be a function from the set X to Y .
If X has more elements that Y then f is not one to one.
If X and Y have the same number of elements and f is ontothen f is one to one.
If X and Y have the same number of elements and f is one toone then f is onto.
T. Singh Introduction
More abstract formulations of the three principles enunciated thusfar are the following.Let f : X → Y be a function from the set X to Y .
If X has more elements that Y then f is not one to one.
If X and Y have the same number of elements and f is ontothen f is one to one.
If X and Y have the same number of elements and f is one toone then f is onto.
T. Singh Introduction
Chinese Remainder Theorem
Theorem
Let m and n be relatively prime positive integers and let a and bbe integers where 0 ≤ a ≤ m− 1 and 0 ≤ b ≤ n− 1. Then there isa positive integer x such that the remainder when x is divided bym is a and the remainder when x is divided by n is b.
T. Singh Introduction
PHP strong form
Theorem
Let q1, q2, · · · , qn be positive integers. If∑n
i=1 qi − n + 1 objectsare put into n boxes, then either the first box contains at least q1objects or the second box contains at least q2 objects, ..., or thenth box contains qn objects.
T. Singh Introduction
PHP contd.
In the elementary mathematics the strong form of PHP is mostoften applied in the special case when q1, q2, · · · , qn are equal tosome integer r .
Theorem
If n(r − 1) + 1 objects are put into n boxes, then at least one ofthe boxes contains r or more of the objects.
Equivalently
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is greaterthan r − 1, then at least one of the integers is greater than orequal to r .
T. Singh Introduction
PHP contd.
In the elementary mathematics the strong form of PHP is mostoften applied in the special case when q1, q2, · · · , qn are equal tosome integer r .
Theorem
If n(r − 1) + 1 objects are put into n boxes, then at least one ofthe boxes contains r or more of the objects.
Equivalently
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is greaterthan r − 1, then at least one of the integers is greater than orequal to r .
T. Singh Introduction
PHP contd.
In the elementary mathematics the strong form of PHP is mostoften applied in the special case when q1, q2, · · · , qn are equal tosome integer r .
Theorem
If n(r − 1) + 1 objects are put into n boxes, then at least one ofthe boxes contains r or more of the objects.
Equivalently
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is greaterthan r − 1, then at least one of the integers is greater than orequal to r .
T. Singh Introduction
PHP contd.
Another average principle is as follows:
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is lessthan r + 1, then at least one of the integers is less than r + 1.
or
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is at leastr , then at least one of the integers m1,m2, · · · ,mn satisfies mi ≥ r .
T. Singh Introduction
PHP contd.
Another average principle is as follows:
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is lessthan r + 1, then at least one of the integers is less than r + 1.
or
Theorem
If the average of n non negative integers m1,m2, · · · ,mn is at leastr , then at least one of the integers m1,m2, · · · ,mn satisfies mi ≥ r .
T. Singh Introduction
Ramsey Theorem
Theorem
Of six(or more) people, either there are three, each pair of whomare acquainted, or there are three, each of whom are unacquainted.
Equivalently
Theorem
K6 → K3,K3 is the assertion that no matter how the edges of K6
are colored with the colors red and blue, there is always amonochromatic triangle.
T. Singh Introduction
Ramsey Theorem
Theorem
Of six(or more) people, either there are three, each pair of whomare acquainted, or there are three, each of whom are unacquainted.
Equivalently
Theorem
K6 → K3,K3 is the assertion that no matter how the edges of K6
are colored with the colors red and blue, there is always amonochromatic triangle.
T. Singh Introduction
Ramsey Theorem
Theorem
Of six(or more) people, either there are three, each pair of whomare acquainted, or there are three, each of whom are unacquainted.
Equivalently
Theorem
K6 → K3,K3 is the assertion that no matter how the edges of K6
are colored with the colors red and blue, there is always amonochromatic triangle.
T. Singh Introduction
Ramsey Theorem contd.
More generally, Ramsey’s theorem can stated as as
Theorem
If m ≥ 2 and n ≥ 2 are integers, then there is a positive integer psuch that Kp → Km,Kn.
The Ramsey number r(m, n) is the smallest positive integer p suchthat Kp → Km,Kn.
T. Singh Introduction
Ramsey Theorem contd.
More generally, Ramsey’s theorem can stated as as
Theorem
If m ≥ 2 and n ≥ 2 are integers, then there is a positive integer psuch that Kp → Km,Kn.
The Ramsey number r(m, n) is the smallest positive integer p suchthat Kp → Km,Kn.
T. Singh Introduction
Ramsey Theorem contd.
More generally, Ramsey’s theorem can stated as as
Theorem
If m ≥ 2 and n ≥ 2 are integers, then there is a positive integer psuch that Kp → Km,Kn.
The Ramsey number r(m, n) is the smallest positive integer p suchthat Kp → Km,Kn.
T. Singh Introduction
Generalized Ramsey Theorem
Theorem
Given integers t ≥ 2 and integers m1,m2, · · · ,mk ≥ t, there existsan integer p such that K t
p → K tm1,K t
m2, . . . ,K t
mk.
The smallest such integer p is the Ramsey numberrt(m1,m2, · · · ,mk) = p.
Remark
Note that when t = 1, then r1(m1,m2, · · · ,mk) is the smallestinteger p such that if the element of set of p elements are coloredwith one of the colors c1, c2, · · · , ck then either there are mi
elements of color ci .So r1(m1,m2, · · · ,mk) =
∑ki=1mi − k + 1.
Which is most generalized form of strong form of PHP.
T. Singh Introduction
Generalized Ramsey Theorem
Theorem
Given integers t ≥ 2 and integers m1,m2, · · · ,mk ≥ t, there existsan integer p such that K t
p → K tm1,K t
m2, . . . ,K t
mk.
The smallest such integer p is the Ramsey numberrt(m1,m2, · · · ,mk) = p.
Remark
Note that when t = 1, then r1(m1,m2, · · · ,mk) is the smallestinteger p such that if the element of set of p elements are coloredwith one of the colors c1, c2, · · · , ck then either there are mi
elements of color ci .So r1(m1,m2, · · · ,mk) =
∑ki=1mi − k + 1.
Which is most generalized form of strong form of PHP.
T. Singh Introduction
Generalized Ramsey Theorem
Theorem
Given integers t ≥ 2 and integers m1,m2, · · · ,mk ≥ t, there existsan integer p such that K t
p → K tm1,K t
m2, . . . ,K t
mk.
The smallest such integer p is the Ramsey numberrt(m1,m2, · · · ,mk) = p.
Remark
Note that when t = 1, then r1(m1,m2, · · · ,mk) is the smallestinteger p such that if the element of set of p elements are coloredwith one of the colors c1, c2, · · · , ck then either there are mi
elements of color ci .So r1(m1,m2, · · · ,mk) =
∑ki=1mi − k + 1.
Which is most generalized form of strong form of PHP.
T. Singh Introduction
Generalized Ramsey Theorem
Theorem
Given integers t ≥ 2 and integers m1,m2, · · · ,mk ≥ t, there existsan integer p such that K t
p → K tm1,K t
m2, . . . ,K t
mk.
The smallest such integer p is the Ramsey numberrt(m1,m2, · · · ,mk) = p.
Remark
Note that when t = 1, then r1(m1,m2, · · · ,mk) is the smallestinteger p such that if the element of set of p elements are coloredwith one of the colors c1, c2, · · · , ck then either there are mi
elements of color ci .So r1(m1,m2, · · · ,mk) =
∑ki=1mi − k + 1.
Which is most generalized form of strong form of PHP.
T. Singh Introduction
Thank you
T. Singh Introduction
