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Given two equations E1 and E2, the disjunctive Rado number for E1 and E2 is the least integer n, provided that it exists, such that for every coloring of the set {1, 2, . . . , n} with two colors there exists a monochromatic solution to either E1 or E2. If no such integer n exists, then the disjunctive Rado number for E1 and E2 is infinite. Let R(c, k)(More)
For integers n ≥ 1 and k ≥ 0, let Mk (n) represent the minimum number of monochromatic solutions to x + y < z over all 2-colorings of {k + 1, k + 2, . . . , k + n} . We show that for any k ≥ 0, Mk (n) = Cn3(1 + ok(1)), where C = 1 12(1+22)2 ≈ .005686. A structural result is also proven, which can be used to determine the exact value of Mk(n) for given k and(More)
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