2,319 research outputs found

    The largest set partitioned by a subfamily of a cover

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    Define [lambda](n) to be the largest integer such that for each set A of size n and cover J of A, there exist B [subset of or equal to] A and G [subset of or equal to] J such that |B| = [lambda](n) and the restriction of G to B is a partition of B. It is shown that when n [ges] 3. The lower bound is proved by a probabilistic method. A related probabilistic algorithm for finding large sets partitioned by a subfamily of a cover is presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28503/1/0000300.pd

    Colouring set families without monochromatic k-chains

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    A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which nn-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to kk-chain-free families. Given a family F\mathcal{F} of subsets of [n][n], we define an (r,k)(r,k)-colouring of F\mathcal{F} to be an rr-colouring of the sets without any monochromatic kk-chains F1⊂F2⊂⋯⊂FkF_1 \subset F_2 \subset \dots \subset F_k. We prove that for nn sufficiently large in terms of kk, the largest kk-chain-free families also maximise the number of (2,k)(2,k)-colourings. We also show that the middle level, ([n]⌊n/2⌋)\binom{[n]}{\lfloor n/2 \rfloor}, maximises the number of (3,2)(3,2)-colourings, and give asymptotic results on the maximum possible number of (r,k)(r,k)-colourings whenever r(k−1)r(k-1) is divisible by three.Comment: 30 pages, final versio

    Quantitative Tverberg, Helly, & Carath\'eodory theorems

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    This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page

    Matchings, coverings, and Castelnuovo-Mumford regularity

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    We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio

    Finding a non-minority ball with majority answers

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    Suppose we are given a set of nn balls {b1,…,bn}\{b_1,\ldots,b_n\} each colored either red or blue in some way unknown to us. To find out some information about the colors, we can query any triple of balls {bi1,bi2,bi3}\{b_{i_1},b_{i_2},b_{i_3}\}. As an answer to such a query we obtain (the index of) a {\em majority ball}, that is, a ball whose color is the same as the color of another ball from the triple. Our goal is to find a {\em non-minority ball}, that is, a ball whose color occurs at least n2\frac n2 times among the nn balls. We show that the minimum number of queries needed to solve this problem is Θ(n)\Theta(n) in the adaptive case and Θ(n3)\Theta(n^3) in the non-adaptive case. We also consider some related problems
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