305 research outputs found
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , 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 -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
-tilings in dense graphs.Comment: 19 pages, to appear in CP
An improved lower bound for Folkman's theorem
Folkman's Theorem asserts that for each , there exists a
natural number such that whenever the elements of are
two-coloured, there exists a set of size with the property
that all the sums of the form , where is a nonempty
subset of , are contained in and have the same colour. In 1989,
Erd\H{o}s and Spencer showed that , where is
an absolute constant; here, we improve this bound significantly by showing that
for all .Comment: 5 pages, Bulletin of the LM
Preparation of small silicon carbide quantum dots by wet chemical etching
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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