The Zarankiewicz problem in 3‐partite graphs

Let be a graph, be an integer, and write for the maximum number of edges in an ‐vertex graph that is ‐partite and has no subgraph isomorphic to . The function has been studied by many researchers. Finding is a special case of the Zarankiewicz problem. We prove an analog of the Kövári‐Sós‐Turán theorem for 3‐partite graphs by showing for . Using Sidon sets constructed by Bose and Chowla, we prove that this upper bound is asymptotically best possible in the case that and is odd, that is, for . In the cases of and , we use a result of Allen, Keevash, Sudakov, and Verstraëte, to show that a similar upper bound holds for all and gives a better constant when . Finally, we point out an interesting connection between difference families from design theory and .     

Turán numbers for Berge-hypergraphs and related extremal problems

Let $F$ be a graph. We say that a hypergraph $H$ is a Berge-$F$ if there is a bijection $f:E\left(F\right)\to E\left(H\right)$ such that $e?f\left(e\right)$ for every $e\in E\left(F\right)$. Note that Berge-$F$ actually denotes a class of hypergraphs. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$F$ is denoted ${\mathrm{ex}}_r(n,\text{Berge-}F)$. In this paper, we investigate the case when $F=K_{s,t}$ and establish an upper-bound when $r\ge 3$, and a lower-bound when $r=4$ and $t$ is large enough compared to $s$. Additionally, we prove a counting result for $r$-graphs of girth five that complements the asymptotic formula ${\text{ex}}_3(n,\text{Berge-}\{C_2,C_3,C_4\})=\frac{1}{6}{n}^{3∕2}+o\left(n^{3∕2}\right)$ of Lazebnik and Verstraëte (2003).     

Star coloring bipartite planar graphs

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009     

MgI2-catalyzed halo aldol reaction: a practical approach to (E)-beta-iodovinyl-beta'-hydroxyketones

A novel generation of 1-iodo-3-siloxy-1,3-butadienes has been developed by reacting trimethylsilyl iodide (TMS-I) with alpha, beta-unsaturated ketones in dichloromethane at 0 degrees C without the use of any catalyst. The halo aldol reaction of these butadiene intermediates with aldehydes was efficiently carried out by using magnesium iodide as the catalyst. Twelve beta-iodo-alpha,beta-unsaturated-beta'-hydroxyketones (halo aldols) have been synthesized under the new condition with excellent geometric selectivity and good chemical yields (>80% chemical yields for 11 examples).     

New asymmetric halo aldol reaction provides a novel approach to biologically important chiral cyclothers and cycloamines

[reaction: see text] A new asymmetric halo aldol reaction has been developed by reacting cyclopropyl carbonyl derived enolates with aldehydes. The absolute structure was unambiguously confirmed by X-ray structural analysis. Eight examples were reported with good yields and up to complete control of diastereomeric excesses. These halo aldol products have been readily cyclized in the presence of weak bases to produce chiral 2,3-disubstituted tetrahydrofuran derivatives in good yield without any observed epimerization.     

Epoxides and aziridines from diazoacetates via ylide intermediates

An effective methodology is reported for stereospecific epoxidation and aziridination via carbonyl ylide intermediates using rhodium(II) acetate catalyzed reactions of phenyl- and styryldiazoacetates with aldehydes, ketones, or imines.     

The kinetics of the reactions of O(3P) atoms with dimethyl ether and methanol

The kinetics of the reaction of O + CH₃OCH₃ were investigated using fast-flow apparatus equipped with ESR and mass-spectrometric detection. The concentration of O(³P) atoms to CH₃OCH₃ was varied over an unusually large range. The rate constant for reaction was found to be k = (5.0 ± 1.0) × 10¹² exp [(?2850 ± 200/RT)] cm³ mole^?1 sec^?1. The reaction O + CH₃OH was studied using ESR detection. Based on an assumed stoichiometry of two oxygen atoms consumed per molecule of CH₃OH which reacts, we obtain a value of k = (1.70 ± 0.66) × 10¹² exp [(?2,280 ± 200/RT)] cm³ mole^?1 sec^?1 for the reaction The results obtained in this study are compared with the results from other workers on these reactions. The observation of essentially equal activation energies in these two reactions is indicative of approximately equal C? H bond strengths in CH₃OCH₃ and CH₃OH. This is in agreement with recent measurements of these bond energies.     

Exploiting nucleotidylyltransferases to prepare sugar nucleotides

[reaction: see text] Enzymatic approaches to prepare sugar nucleotides are gaining in importance and offer several advantages over chemical synthesis including high yields and stereospecificity. We report the cloning, expression, and purification of two new wild-type thymidylyltransferases and observed catalysis with a wide variety of substrates. Significant product inhibition was not observed with the enzymes studied over a 24 h period, enabling the efficient preparation of 15 sugar nucleotides, clearly demonstrating the synthetic utility of these biocatalysts.     

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators

Let L =-?+V(x) be a Schr?dinger operator, where ? is the Laplacian on ■~n,while nonnegative potential V(x) belonging to the reverse H?lder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schr?dinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schr?dinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schr?dinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.