Schur's theorem and numbers

Question

Can you give a proof for bounds of Schur's numbers $S(r)$? Please suggest me articles to have better idea of Schur's theorem (Ramsey theory).

Answer 1

The proof of the Schur's theorem (Ramsey theory) is actually straightforward with an ingenious trick that:

1. Relies on Ramsey's theorem $R(\underset{r}{\underbrace{3,\dots,3}})$ to guarantee that we will find a monochromatic clique $K_3$ of a certain color $c,$ regardless of how the edges are colored between integers $\{1,2,\dots,S\}$ with $S$ an integer (finite). In this set up $r$ is the number of colors $C$ used to partition the integers $\{1,2,\dots,S\}$ - not the edges, but the actual integers, which are now vertices of the graph! -

2. Using this complete flexibility in coloring the edges, and still getting a monochromatic $K_3$ by Ramsey's theorem, we opt for an implicit mapping between the colors of the edges and the colors $r$ that partition the integers.

The trick is to color the edges of $K_S$ in a way that the color $c_i$ of the edge between two integers $i$ and $j$ corresponds to the color $C_i$ assigned to the integer $\vert i - j \vert$ when initially colored with $r$ different colors (partitions).

One thing is guaranteed, which is that in our monochromatic $K_3$ within $K_S$ the edges between the vertices $i,j,$ and $k$ will map to the integers $\vert i-j \vert,$ $\vert j-k \vert,$ and $\vert i-k \vert$ which are all in the same partition of the integers $\{1,2,\dots,S\}$ among $r$ colors.

Therefore, if we label $x=\vert i-j \vert,$ $y=\vert j-k \vert,$ and $z=\vert i-k\vert,$ we will have three integers in the same partition (color), fulfilling the identity $x+y=z.$

Hence, the existence of $S(r)$ is proven, and its bound linked to a Ramsey's number.$\tag*{∎}$