3 research outputs found

    Monochromatic 4-term arithmetic progressions in 2-colorings of Zn\mathbb Z_n

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    This paper is motivated by a recent result of Wolf \cite{wolf} on the minimum number of monochromatic 4-term arithmetic progressions(4-APs, for short) in Zp\Z_p, where pp is a prime number. Wolf proved that there is a 2-coloring of Zp\Z_p with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construction of a 2-coloring with 9.3% fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in Zn\Z_n for general nn. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in all 2-colorings of Zn\Z_n. Wolf proved that any 2-coloring of Zp\Z_p has at least (1/16+o(1))p2(1/16+o(1))p^2 monochromatic 4-APs. We improve this lower bound into (7/96+o(1))p2(7/96+o(1))p^2. Our results on Zn\Z_n naturally apply to the similar problem on [n][n] (i.e., {1,2,...,n}\{1,2,..., n\}). In 2008, Parillo, Robertson, and Saracino \cite{prs} constructed a 2-coloring of [n][n] with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham \cite{BCG} extended their methods and used an extensive computer search to construct a 2-coloring of [n][n] with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of [n][n] with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings.Comment: 23 pages, 4 figure

    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)(x,y,x+ay) whose entries are from the set {1,…,n}\{1,\dots,n\}, subject to a coloring with two different colors. Previously, only asymptotic formulas for such bounds were known, and only for a∈Na\in\mathbb{N}. Using symbolic computation techniques, these results are extended here to arbitrary a∈Ra\in\mathbb{R}. Furthermore, we give exact formulas for the minimum number of monochromatic Schur triples for a=1,2,3,4a=1,2,3,4, and briefly discuss the case 0<a<10<a<1.Comment: 24 Page

    On the monochromatic Schur Triples type problem

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    We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1, n], of monochromatic {x, y, x + ay} triples for a β‰₯ 1. We give a new proof of the original case of a = 1. We show that the minimum number + O(n) when a β‰₯ 2. We also find a new upper bound for the minimum number, over all r-colorings of [1, n], of monochromatic Schur triples, for r β‰₯ 3. of such triples is at most n 2 2a(a 2 +2a+3)
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