69 research outputs found

    Equation-regular sets and the Fox–Kleitman conjecture

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    Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑k i=1 (xi − yi) = b is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of the corresponding equation for all b ≥ 1. In particular, this independently confirms the conjecture for k = 3. We also briefly discuss the case k = 4

    Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations

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    A linear equation is rr-regular, if, for every rr-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoicic conjectured that the equation x1+2x2++2n2xn12n1xn=0x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0, for any n2n \geq 2, has a degree of regularity of n1n-1, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoicic's family of equations indeed have a degree of regularity of n1n-1. We also provide a few extensions of this result.Comment: 8 page

    On the degree of regularity of a certain quadratic Diophantine equation

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    We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic progressions

    The regularity method for graphs with few 4-cycles

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    We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n^(3/2)) edges. * For r ≥ 3, every n-vertex r-graph with girth greater than 5 has o(n^(3/2)) edges. * Every subset of [n] without a nontrivial solution to the equation x₁+x₂+2x₃ = x₄+3x₅ has size o(√n)

    The regularity method for graphs with few 4-cycles

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    We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every nn-vertex graph with no 5-cycle can be made triangle-free by deleting o(n3/2)o(n^{3/2}) edges. * For r3r \geq 3, every nn-vertex rr-graph with girth greater than 55 has o(n3/2)o(n^{3/2}) edges. * Every subset of [n][n] without a nontrivial solution to the equation x1+x2+2x3=x4+3x5x_1 + x_2 + 2x_3 = x_4 + 3x_5 has size o(n)o(\sqrt{n}).Comment: 23 page

    A collection of open problems in celebration of Imre Leader's 60th birthday

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    One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems included cover many of the areas of combinatorial mathematics that Imre is most associated with: including extremal problems on graphs, set systems and permutations, and Ramsey theory. This is a personal selection of problems which we find intriguing and deserving of being better known. It is not intended to be systematic, or to consist of the most significant or difficult questions in any area. Rather, our main aim is to celebrate Imre and his mathematics and to hope that these problems will make him smile. We also hope this collection will be a useful resource for researchers in combinatorics and will stimulate some enjoyable collaborations and beautiful mathematics
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