Consider the following finite version Hindman theorem:
For every sufficiently large $N\in\omega$ and 2-partition of $N=N\_0\cup N\_1$, there are $i<2,a,b,c\in N\_i$ such that $a+b=c$.


The only proof I know for this is by iteratively using Hales-Jewett theorem. What are the alternative proofs?



Finite Hindman theorem
This is Schur's theorem, it follows from Ramsey theorem: consider the complete graph with vertices $1,\ldots,N+1$ and color the edge between vertices $i$ and $j$ with color $s\in \{0,1\}$ iff $|i-j|\in N\_i$. A monochromatic triangles provides a monochromatic solution of $a+b=c$.


