Exact Lower Bounds for Monochromatic Schur Triples and Generalizations
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We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples (x,y,x+ay) whose entries are from the set {1,…,n}, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for a∈ℕ. Using symbolic computation techniques, these results are extended here to arbitrary a∈ℝ. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for a=1,2,3,4, and briefly discuss the case 0<a<1.
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