Given a finite poset P , we consider the largest size La(n, P ) of a family of subsets of [n] := {1, . . . , n} that contains no (weak) subposet P . This problem has been studied intensively in recent years, and it is conjectured that π(P ) := lim n→∞ La(n, P )/ n ⌊ n 2 ⌋ exists for general posets P , and, moreover, it is an integer. For k ≥ 2 let D k denote the k-diamond poset {A < B 1 , . . . , B k < C}. We study the average number of times a random full chain meets a P -free family, called the Lubell function, and use it for P = D k to determine π(D k ) for infinitely many values k. A stubborn open problem is to show that π(D 2 ) = 2; here we make progress by proving π(D 2 ) ≤ 2 3 11 (if it exists).

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Increasing attention is being paid to the study of families of subsets of an n-set that contain no subposet P. Especially, we are interested in such families of maximum size given P and n. For certain P this problem is solved for general n, while for other P it is extremely challenging to find even an approximate solution for large n. It is conjectured that for any P, the maximum size is asymptotic to a constant times, where the constant is a certain integer depending on P. This survey has two purposes. First, we want to bring this exciting line of research to the attention of a wider audience. Second, we want to make experts aware of the broad range of recent progress in the area.

One-dimensional nanostructure materials offer opportunities for improving performance of electrochemical sensors. In this work, vertically ZnO nanorods (ZNRs) sensitized with gold nanoparticles (GNPs) were designed and fabricated onto indium tin oxide coated polyethylene terephthalate (ITO/PET) film for dopamine sensing. ZNRs that helpful for electric signal collecting by providing electron transfer pathways were electrodeposited on ITO/PET film firstly. Then GNPs that possess excellent electrocatalytic activity toward target were decorated onto ZNRs via potentiodynamic electrodeposition. These gold nanoparticles sensitized ZnO nanorods arrays (GNPs/ZNRs) combine the advantages of GNPs and ZNRs, thus providing chance to develop electrochemical sensors with ultrahigh sensitivity and excellent selectivity. Several important nervous system diseases (such as Parkinson's disease, schizophrenia, senile dementia, AIDS, et al.) have proved to be associated with dysfunctions of dopamine system. So, the detection of dopamine becomes essential in clinical medical practice and nerve physiology study. When used for dopamine sensing, the fabricated electrochemical sensor shows two linear dynamic ranges (0.01-20 μM and 50-1000 μM) toward dopamine. Moreover, this proposed electrochemical sensor has been successfully applied to the determination of dopamine in human urine with satisfied recoveries (95.3% to 111.3%) and precision (1.1% to 8.4% of RSD).

Given a finite poset P , let La(n, P ) denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the nset, to give simpler new proofs of the known asymptotic behavior of La(n, P ), as n → ∞, when P is the r-fork V r , the four-element N poset N , and the four-element butterfly-poset B.