An improved lower bound for Folkman's theorem

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