305 research outputs found

    Tilings in randomly perturbed dense graphs

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    A perfect HH-tiling in a graph GG is a collection of vertex-disjoint copies of a graph HH in GG that together cover all the vertices in GG. In this paper we investigate perfect HH-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds mm random edges to it. Specifically, for any fixed graph HH, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect HH-tiling with high probability. Our proof utilises Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect HH-tilings in dense graphs.Comment: 19 pages, to appear in CP

    An improved lower bound for Folkman's theorem

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    Folkman's Theorem asserts that for each kNk \in \mathbb{N}, there exists a natural number n=F(k)n = F(k) such that whenever the elements of [n][n] are two-coloured, there exists a set A[n]A \subset [n] of size kk with the property that all the sums of the form xBx\sum_{x \in B} x, where BB is a nonempty subset of AA, are contained in [n][n] and have the same colour. In 1989, Erd\H{o}s and Spencer showed that F(k)2ck2/logkF(k) \ge 2^{ck^2/ \log k}, where c>0c >0 is an absolute constant; here, we improve this bound significantly by showing that F(k)22k1/kF(k) \ge 2^{2^{k-1}/k} for all kNk\in \mathbb{N}.Comment: 5 pages, Bulletin of the LM

    Preparation of small silicon carbide quantum dots by wet chemical etching

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    Fabrication of nanosized silicon carbide (SiC) crystals is a crucial step in many biomedical applications. Here we report an effective fabrication method of SiC nanocrystals based on simple electroless wet chemical etching of crystalline cubic SiC. Comparing an open reaction system with a closed reaction chamber, we found that the latter produces smaller nanoparticles (less than 8 nm diameter) with higher yield. Our samples show strong violet-blue emission in the 410–450 nm region depending on the solvents used and the size. Infrared measurements unraveled that the surface of the fabricated nanoparticles is rich in oxidized carbon. This may open an opportunity to use standard chemistry methods for further biological functionalization of such nanoparticles
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