Chapter 15 A proof without words that \(R(a,b,c)\) exists
Project by: Chenye Hua, Kai Hang Mok, Brian Tran, Jackson Voong, and Ni Wang.
\(\textbf{Summary:}\) Our project is a proof without words.
We are proving the existence of the Ramsey number \(R(a,b,c)\) based on the fact that \(R(a,b)\) exists for all positive integers \(a,b\text{.}\)
Let our colours be red, blue, and green. We re–colour the red and blue elements light blue. The idea is to choose \(M\) such that \(M=R(d,c)\) where \(d=R(a,b)\text{.}\)
If \(K_M\) contains a green \(K_c\text{,}\) then we are done.
If it does not contain a green \(K_c\text{,}\) we have a light blue \(K_d\) such as \(d=R(a,b)\) because we carefully chose our \(M\text{.}\)
We now look at the original red and blue edge–colouring of the chosen ('light blue') \(K_d=K_{R(a,b)}\) . Then we either have a red \(K_a\) or a blue \(K_b\text{.}\)
See Figure 15.0.1.
Watch a visual proof that \(R(a,b,c)\) exists through the link below.
Enjoy!