A 2-Coloring of [1, N] Can Have (1/22)N2+O(N) Monochromatic Schur Triples, But Not less!
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We prove the statement of the title, thereby solving a $100 problem of Ron Graham. This was solved independently by Tomasz Schoen.
[1] Tomasz Schoen. The Number of Monochromatic Schur Triples , 1999, Eur. J. Comb..
[2] Vojtech Rödl,et al. On Schur Properties of Random Subsets of Integers , 1996 .