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Ramsey numbers

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Season 2 • Episode 20 • Ramsey numbers

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Ramsey numbers

Season 2

Episode 20 Time frame 2 periods

Prerequisites :

Basinotionsandvoabularyaboutgraphs.

Objectives :

•

Introdue the onept of a omplete graph.

•

Modelize a problemusing graphs.

•

Disover the onept of Ramsey numbers.

Materials :

•

Beamer.

1 – A party problem 1 period

The following problemisintrodued:

You're organising a party. Some of the guests will know eah other, while

otherswon't. Forthe sakeof simpliity,weadmitthattherelationofknowing

eah other issymmetri.

What is the least number of guest to invite so that at least

m

people will

mutually know eah other,or atleast

n

people willbe omplete strangers?

Some simple examplesare studiedwith the wholelass :

• m = 2

and

n = 2

;

• m = 3

and

n = 2

;

• m = 2

and

n = 3

;

• m = 2

and

n ∈

N;

Then, students have to stand up and work in groups in the problem with

m = 3

and

n = 3

.

If there isenough time, the problemwith

m = 4

and

n = 3

an be takledtoo.

2 – Turning it into a graph problem 1 period

The lass isasked to modelizethis problem with the use of graph.

Hintsan begiven by the teaher :

•

What willbethe verties of the graph?

•

How tosymbolize the fat the people willknoweah other ornot?

The onepts of omplete graph and Ramsey number are introdued, along with the

oial notations

K n

and

R(m, n)

. A quik biography of Ramsey isgiven.

The proof of the fatthat

R(3, 3) = 6

is given.

Références

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