CO8 Small Ramsey Numbers: r(3, 4)=9, r(3, 5)= 14, r(4,4) =18

Proofs for the Ramsey Numbers r(3, 4), r(3,5), and r(4,4). One proposition, in one sweep, helps us find all three. Subscribe ‪@Shahriari‬ for more undergraduate math videos. After quickly reviewing Ramsey's Theorem and Ramsey Numbers, we prove that if r(n-1,m) and r(n,m-1) are both even integers, then r(n,m) is less than or equal to r(n-1,m)+r(n,m-1)-1. This allows us to show that r(3,4) is no more than 9, and, then it follows that r(3, 5) is less than or equal to 14 and r(4,4) is less than or equal to 18. Examples are provided to show that we indeed have equality.#combinatorics 00:00 Introduction 00:55 Reminder r(3,3)=6 (   • CO6 What is Ramsey Theory?  ) 01:34 r(3,4) = 9. What does it mean? 04:41 The arrow notation 06:27 Ramsey's Theorem (   • CO7 Proof of Ramsey's Theorem for 2 colors  ) 07:22 Ramsey Numbers 08:01 Ramsey Numbers: Initial cases 09:03 Recursive inequality for Ramsey Numbers (   • CO7 Proof of Ramsey's Theorem for 2 colors  ) 10:48 Key fact & Strategy for Proof of small Ramsey numbers 13:46 Proof of Key Fact split into 3 cases 16:52 Proof of Cases 1 & 2 19:16 Proof of Case 3 23:01 Applying the Key Fact to get upper bounds for r(3,4), r(3, 5), and r(4,4) 23:46 Example: r(3,4) is more than 8 26:04 Example: r(3,5) is more than 13 27:46 Example: r(4,4) is more than 17 28:58 Other videos of interest Next Ramsey Theory Video:    • CO21 Upper and Lower Bounds for Ramsey Num...   Videos on Ramsey Theory: What is Ramsey Theory?    • CO6 What is Ramsey Theory?   Proof of Ramsey's Theorem for 2 colors    • CO7 Proof of Ramsey's Theorem for 2 colors   Small Ramsey Numbers r(3,4)=9, r(3,5)=14, and r(4,4) = 18    • CO8 Small Ramsey Numbers: r(3, 4)=9, r(3, ...   Upper & Lower Bounds for Ramsey Numbers    • CO21 Upper and Lower Bounds for Ramsey Num...   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: https://doi.org/10.1017/9781108568708 For an annotated list of available videos for Combinatorics see https://pomona.box.com/s/by2ay2872avx... YouTube Playlist:    • Combinatorics, An Invitation   Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award. Subscribe to my channel ‪@Shahriari‬​