On Monochromatic Pairs of Nondecreasing Diameters
نویسندگان
چکیده
منابع مشابه
On four color monochromatic sets with nondecreasing diameter
Let m and r be positive integers. Define f(m, r) to be the least positive integer N such that for every coloring of the integers 1, . . . , N with r colors there exist monochromatic subsets B1 and B2 (not necessarily of the same color), each having m elements, such that (a) max(B1)−min(B1) ≤ max(B2)−min(B2), and (b) max(B1) < min(B2). We improve previous upper bounds to determine that f(m, 4) =...
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Let g(m, t) denote the minimum integer s such that for every 2-coloring of the integers in the interval [1, s], there exist t subsets A1, A2, . . . , At, of size m satisfying: (i) Ai for every i = 1, 2, . . . , t is monochromatic (not necessarily the same color) (ii) max(Ai) ≤ min(Ai+1) for every i = 1, 2, . . . , t − 1, and (iii) either diam(Ai) ≤ diam(Ai+1) for every i = 1, 2, . . . , t − 1 o...
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| We consider the following problem: Let r be a n-bead rosary with m white beads and n ? m black beads. Let t be an integer, t n. Denote by MC t (r) the number of pairs, of monochromatic beads which are within distance t apart, in the rosary r. What is the minimum value of MC t (), when the minimum is taken over all n-bead rosaries which consists of m white beads and n ? m black beads? We prove...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2019
ISSN: 1077-8926
DOI: 10.37236/8003