Introduction to Ramsey Theory: Students' Projects

We will prove by contradiction, by providing a 3–colouring that has no monochromatic two terms with difference in \(F_E\text{.}\)

Here is some notation for the proof: For \(x\in\mathbb{R}\text{,}\) \(\{x\}\) is the fractional part of \(x\) defined as \(\{x\}=x-\lfloor x\rfloor\text{.}\)

Next, the norm of \(x\) is defined as, \(\| x\|=\min(\{x \},1-\{x\}])\text{,}\) so \(\| x\|\) is the shortest distance from \(x\) to an integer. By Applying Binet's formula we get:

Now we will show that \(\lim_{n\to\infty}\|{\varphi\over2}f_{3n}=\frac{1}{2}\text{:}\)

which implies \(\lim_{n\to\infty}\|{\varphi\over2}f_{3n}\|= {1\over2}\text{.}\)

Additionally \(\frac{1}{3}\lt 0.382=\|{\varphi\over2}f_3\|\leq\|{\varphi\over2}f_{3n}\|\text{.}\)

We define a \(3\)–colouring \(\Psi:\mathbb{Z}^+\rightarrow\{1,2,3\}\) in the following way: For \(C_1=[0,{1\over3})\text{,}\) \(C_2=[{1\over3},{2\over3})\text{,}\) and \(C_3=[{2\over3},1)\text{,}\) \(\Psi(i)=j\) if and only if \(\left\{\frac{\varphi i}{2}\right\}\in C_j\text{.}\)

Now we show that that \(F_E\) is not 3–accessible by letting \(a\) and \(f\) be integers with \(f\in F_E\text{:}\)

Thus \(\Psi(a)\not= \Psi(a+f)\) because \(a\) and \(a+f\) are in distinct \(C_i\)'s.