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Abstract 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: $r=7/5$ .

Biomanufacturing metal/metallic nanomaterials with ordered micro/nanostructures and controllable functions is of great importance in both fundamental studies and practical applications due to their low toxicity, lower pollution production, and energy conservation. Microorganisms, as efficient biofactories, have a significant ability to biomineralize and bioreduce metal ions that can be obtained as nanocrystals of varying morphologies and sizes. The development of nanoparticle biosynthesis maximizes the safety and sustainability of the nanoparticle preparation. Significant efforts and progress have been made to develop new green and environmentally friendly methods for biocompatible metal/metallic nanomaterials. In this review, we mainly focus on the microbial biomanufacture of different metal/metallic nanomaterials due to their unique advantages of wide availability, environmental acceptability, low cost, and circular sustainability. Specifically, we summarize recent and important advances in the synthesis strategies and mechanisms for different types of metal/metallic nanomaterials using different microorganisms. Finally, we highlight the current challenges and future research directions in this growing multidisciplinary field of biomaterials science, nanoscience, and nanobiotechnology.

We have performed combined elastic neutron diffuse, electrical transport, specific heat, and thermal conductivity measurements on the quasi–one-dimensional Ba 3 Co 2 O 6 (CO 3 ) 0.7 single crystal to characterize its transport properties. A modulated superstructure of polyatomic CO 3 2− is formed, which not only interferes the electronic properties of this compound, but also reduces the thermal conductivity along the c-axis. Furthermore, a large magnetic entropy is observed to be contributed to the heat conduction. Our investigations reveal the influence of both structural and magnetic effects on its transport properties and suggest a theoretical improvement on the thermoelectric materials by building up superlattice with conducting ionic group.

Abstract In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach: 1. Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is $2$-connected and non-bipartite, then it contains cycles of all lengths modulo $k$. 2. For all $k\geq 3$, every $k$-connected graph contains a cycle of length zero modulo $k$. 3. Every $3$-connected non-bipartite graph with minimum degree at least $k+1$ contains $k$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $k$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are bestmore »possible.« less

For integers $n\ge 0$, an iterated triangulation $\mathrm{Tr}(n)$ is defined recursively as follows: $\mathrm{Tr}(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $\mathrm{Tr}(n)$ is the plane triangulation obtained from the plane triangulation $\mathrm{Tr}(n-1)$ by, for each inner face $F$ of $\mathrm{Tr}(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$. In this paper, we show that there exists a 2-edge-coloring of $\mathrm{Tr}(n)$ such that $\mathrm{Tr}(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two questions asked by Axenovich et al. is negative. We also determine the radius 2 graphs $H$ for which there exists $n$ such that every 2-edge-coloring of $\mathrm{Tr}(n)$ contains a monochromatic copy of $H$, extending a result of Axenovich et al. for radius 2 trees.

Abstract Spin-orbit coupled honeycomb magnets with the Kitaev interaction have received a lot of attention due to their potential of hosting exotic quantum states including quantum spin liquids. Thus far, the most studied Kitaev systems are 4 d /5 d -based honeycomb magnets. Recent theoretical studies predicted that 3 d -based honeycomb magnets, including Na 2 Co 2 TeO 6 (NCTO), could also be a potential Kitaev system. Here, we have used a combination of heat capacity, magnetization, electron spin resonance measurements alongside inelastic neutron scattering (INS) to study NCTO’s quantum magnetism, and we have found a field-induced spin disordered state in an applied magnetic field range of 7.5 T <  B (⊥ b -axis) < 10.5 T. The INS spectra were also simulated to tentatively extract the exchange interactions. As a 3 d -magnet with a field-induced disordered state on an effective spin-1/2 honeycomb lattice, NCTO expands the Kitaev model to 3 d compounds, promoting further interests on the spin-orbital effect in quantum magnets.