Some two color, four variable Rado numbers
نویسندگان
چکیده
منابع مشابه
Two Color Off-diagonal Rado-type Numbers
We show that for any two linear homogeneous equations E0, E1, each with at least three variables and coefficients not all the same sign, any 2-coloring of Z+ admits monochromatic solutions of color 0 to E0 or monochromatic solutions of color 1 to E1. We define the 2-color off-diagonal Rado number RR(E0, E1) to be the smallest N such that [1, N ] must admit such solutions. We determine a lower b...
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For positive integers a1, a2, . . . , am, we determine the least positive integer R(a1, . . . , am) such that for every 2-coloring of the set [1, n] = {1, . . . , n} with n > R(a1, . . . , am) there exists a monochromatic solution to the equation a1x1 + . . .+amxm = x0 with x0, . . . , xm ∈ [1, n]. The precise value of R(a1, . . . , am) is shown to be av + v − a, where a = min{a1, a2, . . . , a...
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The 2-color Rado number for the equation x1 + x2 − 2x3 = c, which for each constant c ∈ Z we denote by S1(c), is the least integer, if it exists, such that every 2-coloring, ∆ : [1, S1(c)]→ {0, 1}, of the natural numbers admits a monochromatic solution to x1 +x2−2x3 = c, and otherwise S1(c) = ∞. We determine the 2-color Rado number for the equation x1 + x2 − 2x3 = c, when additional inequality ...
متن کاملDisjunctive Rado numbers
If L1 and L2 are linear equations, then the disjunctive Rado number of the set {L1, L2} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1, 2, . . . , n} there exists a monochromatic solution to either L1 or L2. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers a 1 and b ...
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In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by ErdSs and Rado, are given. These yield improvements over the known bounds for the arising Erd6s-Rado numbers ER(k; l), where the numbers ER(k; l) are defined as the least positive integer n such that for every partition of the k-element subsets of a totally ordered n-element set X into an arbitrary num...
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2008
ISSN: 0196-8858
DOI: 10.1016/j.aam.2007.06.002