I am stuck on a problem from A Course in Combinatorics by Lint and Wilson:
Problem 3F: "Show that for any $c\in \mathbb{N} $, there is a number $N(c)$ with the following property. If $n\geq N(c)$ and the integers in $I= \{ 1,2,...,n\} $ are colored with $c$ colors, then there are three elements, $x,y,z\in I $ (not necessarily distinct), with the same color so that $x+y=z $."
The only idea I have:
If we can show an upper bound for $N(c)$ for a given $c$, then $N(c)$ exists by the well-orderedness of $\mathbb{N} $. Could we compare it to a Ramsey number somehow? Like perhaps $$N(c) \leq R(3,3,...,3;2) \quad ,$$ with $c$ $3'$s in the argument?
We have $\leq $ since the our triple we are trying to find doesn't have to have all distinct components?
Any suggestion would be helpful. Thank you!
Given a coloring of $\{1,2,\dots,n\}$, consider the complete graph with vertices $v_1, v_2, \dots, v_n$ where we give edge $v_i v_j$ the color of $|i-j|$. When is a triangle in this graph monochromatic?