CO6 What is Ramsey Theory?

What are Ramsey numbers and what is Ramsey's Theorem? The example (with proof) of r(3, 3) = 6 is given and other Ramsey numbers discussed. Subscribe @Shahriari for more undergraduate math videos.
00:00 Introduction
00:18 Motivating Problem
02:19 Reformulation using graphs
08:09 The Arrow Notation
09:25 Proposition: If the edges of a K_6 are arbitrarily colored using two colors, there will be at least one monochromatic triangle
09:55 Proof of Proposition on K_6
13:00 Challenge: Can you prove that there are two monochromatic triangles in a 2-coloring of the edges of a K_6?
13:23 Proposition: Possible to color edges of a K_5 with 2 colors and have no monochromatic triangles
14:26 Proof of Proposition on K_5
15:44 Notation: Ramsey number r(3, 3)= 6
17:01 Frank Ramsey, a historical note
17:40 Discussion: Ramsey Theory's main idea
18:36 What does r(4,4) mean?
21:05 More general questions
22:22 The Arrow Notation for two colors
23:09 Ramsey's Theorem
24:15 Ramsey Numbers
24:54 r(3,3)=6; r(1, m) =1; and r(2, m) = m
27:13 Discussion: Subtle point about the arrow notation
29:07 Discussion: What is r(5, 5)? or r(6,6)?
33:39 Looking Ahead
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A series of lectures on introductory Combinatorics. This full course is based on my book
Shahriar Shahriari, An Invitation to Combinatorics, Cambridge University Press, 2022.
DOI: doi.org/10.1017/9781108568708
For an annotated list of available videos for Combinatorics see
pomona.box.com/s/by2ay2872avx...
TH-cam Playlist: • Combinatorics, An Invi...
Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College, in Claremont, CA, U.S.A.

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Thank you so much sir ! Great presentation of Content and Explanation was boom🤯