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

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