A proof without words that \(R(a,b,c)\) exists

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Chapter 25 A proof without words that $R(a,b,c)$ exists

Project by: Chenye Hua, Kai Hang Mok, Brian Tran, Jackson Voong, and Ni Wang.

Summary:

$\textbf{Summary:}$ Our project is a proof without words.

We are proving the existence of the Ramsey number

R (a, b, c)

$R(a,b,c)$ based on the fact that

R (a, b)

$R(a,b)$ exists for all positive integers

a, b .

$a,b\text{.}$

Let our colours be red, blue, and green. We re–colour the red and blue elements light blue. The idea is to choose

M

$M$ such that

M = R (d, c)

$M=R(d,c)$ where

d = R (a, b) .

$d=R(a,b)\text{.}$

If

K_{M}

$K_M$ contains a green

K_{c},

$K_c\text{,}$ then we are done.

If it does not contain a green

K_{c},

$K_c\text{,}$ we have a light blue

K_{d}

$K_d$ such as

d = R (a, b)

$d=R(a,b)$ because we carefully chose our

M .

$M\text{.}$

We now look at the original red and blue edge–colouring of the chosen ('light blue')

K_{d} = K_{R (a, b)}

$K_d=K_{R(a,b)}$ . Then we either have a red

K_{a}

$K_a$ or a blue

K_{b} .

$K_b\text{.}$

See Figure 25.0.1.

🔗
Figure 25.0.1. The idea of our visual proof

Watch a visual proof that

R (a, b, c)

$R(a,b,c)$ exists through the link below.

Enjoy!