t-color selectivity Rado numbers.

摘要

For any t ∈ N , let n = Rs( t) be the least integer n, such that for every t-coloring of the set [1, n], one of the following two cases will occur: (a) There exists a solution (x 1, x2, x3) to the Schur Equation x1 + x 2 = x3, such that Δ(x 1) = Δ(x2) = Δ(x 3). (This is known as a monochromatic solution.) (b) There exists a solution (x1, x2, x3) to the Schur Equation such that Δ(x i) ≠ Δ(xj) for every i, j where 1 ≤ i < j ≤ 3. (This is known as a totally multicolored solution.).;The number n = Rs( t) is known as the t-color selectivity Rado Number. The main results section will show that for every t ∈ N , Rst= 5t2 forteven 2˙5t-12f ortodd Furthermore, if t is even, there exists a unique coloring of maximal length. If t is odd, there exist t+12 different colorings of maximal length.