59 research outputs found
Collinear subsets of lattice point sequencesâAn analog of SzemerĂ©di's theorem
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, âŠ, um is a sequence of plane lattice points with âi=1mâŠui â uiâ1⊠⩜ 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
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
Subspaces of tensors with high analytic rank
It is shown that for any subspace VâFnĂâŻĂnp of d-tensors, if dim(V)â„tndâ1, then there is subspace WâV of dimension at least t/(dr)â1 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 graphsâCoordinated and unstable noâwait schedules
A proper edgeâcoloring 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 of a graph is the smallest number of interval colorable graphs edgeâdecomposing . We prove that for any graph on n vertices. This improves the previously known bound of , 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 such that the graph has an interval coloring using colors. Interval colorings of bipartite graphs naturally correspond to noâwait schedules, say for parentâteacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with participants can be coordinated in noâwait periods. In addition, we show that for any integers and , there is a set of pairs of parents and teachers wanting to talk to each other, such that any noâwait schedules are unstableâthey could last hours and could last hours, but there is no possible noâwait schedule lasting hours if
Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry
A szerzĆ nem jĂĄrult hozzĂĄ nyilatkozatĂĄban a dolgozat nyilvĂĄnossĂĄgra hozĂĄsĂĄhoz
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