Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form $w(z) = \exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential of a compactly supported positive measure $\mu$. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials $U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights $w(z)$ supported on a compact domain.Comment: CRM preprint, 28 pages. To appear in: Complex Analysis and Operator Theory (Special volume in honour of Bjorn Gustafsson). Two references added (Oct. 2, 2008

The number of the maximal triangle-free graphs

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among others the Ruzsa-Szemer\'{e}di triangle removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.Comment: 7 pages. This article is slightly different from the journal version, as it contains comments on more recent development