A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such that $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$ , for integers $k\geq 2$ . In this paper, we add a new form of rationals for which the conjecture is true: $2-2/(2k+1)$ , for $k\geq 2$ . This in turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits $^{\prime}$ s conjecture, where one replaces a single graph by a finite family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits $^{\prime}$ s conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon $^{\prime}$ s conjecture. We also prove an upper bound on the Turán number of theta graphs in an asymmetric setting and employ this result to obtain another new rational exponent for Turán exponents: $r=7/5$ .

In the present work, Pt nanoparticles were produced from a reaction mixture containing a trace amount of cobalt carbonyl salt acting as a shape inducer. Nanoparticle shape evolution during reaction mixture reflux was monitored by characterizing particles extracted from the reaction mixture at different times. It was observed that 5 min of reflux produced spherical nanoparticles, 30 min of reflux produced cube shaped nanoparticles, and 60 min of reflux produced truncated octahedron morphology nanoparticles. It is illustrated that during nanoparticle synthesis the reflux process can provide energy needed for shape transformation from a metastable cube morphology to a truncated octahedron morphology which is thermodynamically the most stable geometry for fcc crystals. An optimization of the reaction reflux is thus needed for isolating metastable shapes.