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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?

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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 $a+b=c$.

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    $\begingroup$ See sfu.ca/~vjungic/RamseyNotes/sec_SchurThm.html for a nicely illustrated proof. $\endgroup$
    – Seva
    Commented Mar 17, 2022 at 9:13
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    $\begingroup$ Strictly speaking, I think another answer to the question as posed is "check two-colourings of $\{1,2,3,4,5\}$ by hand''. Certainly this gives a more effective bound than iterative applications of the HJ theorem. $\endgroup$
    – Ben Green
    Commented Mar 17, 2022 at 10:15

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