Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

Obscured Binary Quasar Cores in SDSS J104807.74+005543.5?

We report the discovery of a possible close binary system of quasars in SDSS J1048+0055. The [OIII]4959,5007 emission lines are clearly double-peaked, and two discrete radio sources with a projected physical separation of ~20 pc are found in the VLBA milliarcsec resolution image at 8.4 GHz. Each of the [O III]4959,5007 doublets and Hbeta can be well modelled by two Gaussians and the line ratio, [O III]5007/Hbeta ~7, is typical of Seyfert 2 galaxies. No broad component of Hbeta was detected and its [O III]5007 luminosity, L_[OIII] ~ 9.2 times 10^42 erg s^-1, is comparable to luminous quasars and is a few ten times more luminous than typical Seyfert galaxies. One natural interpretation is that SDSS J1048+0055 contains two close quasar-like nuclei and the BLR around them are obscured. Other possible models are also discussed. We suggest that double-peaked narrow emission line profile may be an effective way of selecting candidates of binary black holes with intermediate separation

The multiple effects of gradient coupling on network synchronization

Recent studies have shown that synchronizability of complex networks can be significantly improved by asymmetric couplings, and increase of coupling gradient is always in favor of network synchronization. Here we argue and demonstrate that, for typical complex networks, there usually exists an optimal coupling gradient under which the maximum network synchronizability is achieved. After this optimal value, increase of coupling gradient could deteriorate synchronization. We attribute the suppression of network synchronization at large gradient to the phenomenon of network breaking, and find that, in comparing with sparsely connected homogeneous networks, densely connected heterogeneous networks have the superiority of adopting large gradient. The findings are supported by indirect simulations of eigenvalue analysis and direct simulations of coupled nonidentical oscillator networks.Comment: 4 pages, 4 figure