# Find $n$ s.t. if the numbers $1,...,n$ are drawn in three different colours, then there exist $x,y, z$ (not necessarily distinct) so that $x+y=z$.

The question is about Exercise (b).

Exercise (a). Let $$k,l$$ and $$m$$ be positive integer. Show that $$r(k,l,m)$$ exists and that $$r(k,l,m)\leq r(k,r(m,l))$$.

Here $$r(k,l,m)$$ is Ramsey’s number over three colours, so that when colouring a graph $$K_c$$, there exists either a blue $$K_k$$, a red $$K_l$$ or green $$K_m$$.

Exercise (b). Find a value of $$n$$ such that if the numbers $$1,2,...,n-1,n$$ are drawn in three different colours, then there exist $$x,y$$ and $$z$$ (not necessarily distinct) so that $$x+y=z$$.

Solution for (a). By Ramsey's theorem, we know that $$r(l,m)=n$$ exists for complete graphs $$K_{n'\geq n}$$ that consist of edges coloured in red and green. Since $$n\in \Bbb Z^+$$, by Ramsey's theorem, we conclude that $$r(k,r(l,m))=r(k,n)=c$$ exists for complete graphs $$K_{c'\geq c}$$ that consist of edges coloured in blue, red and green, where red and green are considered one colour. Thus, Let a complete graph $$K_c$$ be coloured in blue, red and green. From the existence of $$r(k,r(l,m))$$, we have a blue $$K_k$$ in $$K_c$$ or a red and green $$K_n$$ in $$K_c$$. If we have a blue $$K_k$$, we are done; otherwise, we obtain a red and green $$K_n$$ that contains either a red $$K_l$$ or a green $$K_m$$ from the fact that $$r(l,m)=n$$. That proves the existence of $$r(k,l,m)$$. Therefore, $$r(k,l,m)\leq c=r(k,r(l,m))$$ $$\therefore r(k,l,m)\leq r(k,r(l,m)).$$

Now to section (b) – I have been trying for hours to connect the colouring of the consecutive number $$1,2,...,n-1,n$$ with the result from section (a), but with no success. Now I am trying to find a connection between the sum $$x+y$$ with $$r(l,m)$$ from the inequality $$r(k,l,m)\leq r(k,r(l,m))$$, with the possibility that $$m=l$$, but I have not got an idea how it relates to existence of a number $$z$$ from the set $$\{1,2,...,n-1,n\}$$, so that $$x+y=z$$.

Any help would be appreciated.

• @Shaun Ok. Thank you. Jun 1, 2022 at 19:33
• hi Lord Mansfield. what does $r(k,l,m)$ denote? Jun 1, 2022 at 19:44
• Did you mean not necessarily distinct? Jun 1, 2022 at 19:55
• Saying they're "unnecessarily distinct" says they are distinct but need not have been. Surely you meant "not necessarily distinct". Jun 1, 2022 at 20:02
• r(k,l,m) is Ramsey’s number over three colours, so that when colouring a graph $K_c$, there exists either a blue $K_k$, a red $K_l$ or a green $K_m$. Jun 1, 2022 at 20:51

Let $$n=r(3,3,3)-1$$. Given a colouring of the numbers $$1,2,3,\dots,n$$, colour the edges of the complete graph on the vertex set $$\{0,1,2,\dots,n\}$$ by giving the edge $$\{i,j\}$$ the colour of the number $$|i-j|$$. If there is a monochromatic triangle with vertices $$i\lt j\lt k$$, then $$x=j-i$$, $$y=k-j$$, $$z=k-i$$ is the desired solution.