59 research outputs found

    Collinear subsets of lattice point sequences—An analog of SzemerĂ©di's theorem

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    AbstractSzemerĂ©di's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n â©Ÿ n(k, B) and 0 < a1 < 
 < an is a sequence of integers with an â©œ Bn, then some k of the ai form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m â©Ÿ m(k, B) and u0, u1, 
, um is a sequence of plane lattice points with ∑i=1m
ui − ui−1
 â©œ Bm, then some k of the ui are collinear. Our result, while similar to SzemerĂ©di's theorem, does not appear to imply it, nor does SzemerĂ©di's theorem appear to imply our result

    Formulating Szemerédi's theorem in terms of ultrafilters

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    Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemerédi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length

    M 584.01: Topics in Combinatorics and Optimization

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    Subspaces of tensors with high analytic rank

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    It is shown that for any subspace V⊆Fn×⋯×np of d-tensors, if dim(V)≄tnd−1, then there is subspace W⊆V of dimension at least t/(dr)−1 whose nonzero elements all have analytic rank Ωd,p(r). As an application, we generalize a result of Altman on SzemerĂ©di's theorem with random differences

    Interval colorings of graphs—Coordinated and unstable no‐wait schedules

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    A proper edge‐coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval, that is, form a set of consecutive integers. The interval coloring thickness ξ(G)ξ (G) of a graph GG is the smallest number of interval colorable graphs edge‐decomposing GG . We prove that ξ(G)=o(n)ξ (G) = o (n) for any graph GG on n vertices. This improves the previously known bound of 2[n/5]2 [n/5], see Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with an interval coloring thickness strictly greater than 2, we construct bipartite graphs whose interval coloring spectrum has arbitrarily many arbitrarily large gaps. Here, an interval coloring spectrum of a graph is the set of all integers tt such that the graph has an interval coloring using tt colors. Interval colorings of bipartite graphs naturally correspond to no‐wait schedules, say for parent–teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with nn participants can be coordinated in o(n)o (n) no‐wait periods. In addition, we show that for any integers tt and T,t<TT,t <T , there is a set of pairs of parents and teachers wanting to talk to each other, such that any no‐wait schedules are unstable—they could last tt hours and could last TT hours, but there is no possible no‐wait schedule lasting xx hours if t<x<Tt<x<T

    Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry

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    A szerzƑ nem járult hozzá nyilatkozatában a dolgozat nyilvánosságra hozásához
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