9,814 research outputs found

    On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

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    We study the function M(n,k)M(n,k) which denotes the number of maximal kk-uniform intersecting families F([n]k)F\subseteq \binom{[n]}{k}. Improving a bound of Balogh at al. on M(n,k)M(n,k), we determine the order of magnitude of logM(n,k)\log M(n,k) by proving that for any fixed kk, M(n,k)=nΘ((2kk))M(n,k) =n^{\Theta(\binom{2k}{k})} holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.Comment: 11 page

    Vertex covers by monochromatic pieces - A survey of results and problems

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    This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

    On the Complexity of Anchored Rectangle Packing

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    Nonnegative k-sums, fractional covers, and probability of small deviations

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    More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers n,kn, k satisfying n4kn \geq 4k, every set of nn real numbers with nonnegative sum has at least (n1k1)\binom{n-1}{k-1} kk-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n33k2n \geq 33k^2. This substantially improves the best previously known exponential lower bound neckloglogkn \geq e^{ck \log\log k}. In addition we prove a tight stability result showing that for every kk and all sufficiently large nn, every set of nn reals with a nonnegative sum that does not contain a member whose sum with any other k1k-1 members is nonnegative, contains at least (n1k1)+(nk1k1)1\binom{n-1}{k-1}+\binom{n-k-1}{k-1}-1 subsets of cardinality kk with nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde
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