Search for: All records

Abstract Sidorenko’s conjecture states that, for all bipartite graphs $$H$$, quasirandom graphs contain asymptotically the minimum number of copies of $$H$$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $$r$$-partite $$r$$-uniform hypergraph $$H$$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $$\textrm{ex}(n,H)$$, the maximum number of edges in an $$n$$-vertex $$H$$-free $$r$$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $$3$$-partite $$3$$-uniform tight cycles. 

Chiral oxygen-containing heterocyclic compounds are of great interest for the development of pharmaceuticals. Monoterpenes and their derivatives are naturally abundant precursors of novel synthetic chiral oxygen-containing heterocyclic compounds. In this study, acid catalyzed reactions of salicylic aldehydes with (−)-8-acetoxy-6-hydroxymethyllimonene, readily accessible from α-pinene, leads to the formation of chiral polycyclic products of various structural types. Three of the six isolated chiral heterocyclic products obtained from salicylic aldehyde contain previously unknown polycyclic ring types. Having carried out the reaction in the presence of Brønsted or Lewis acids (Amberlyst 15, trifluoromethanesulfonic acid, trifluoroacetic acid and boron trifluoride etherate) or aluminosilicates (montmorillonite K10, halloysite nanotubes), we found that the nature of products depends on the catalyst as well as the reaction conditions (reaction time, reactant ratio, presence or absence of solvent). Detailed mechanistic insight on the complex cascade reactions for product formation is provided with extensive experimental and quantum mechanical computational studies.