Complete r-partite subgraphs of dense r-graphs

We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964.Comment: 7 page

On the Value of R=\Gamma_h/\Gamma_l at LEP

We show that the present experimental LEP average R=\Gamma_h/\Gamma_l= 20.795 +- 0.040 is not unambiguous due to the presence of substantial systematic effects which cannot be interpreted within gaussian statistics. We find by Montecarlo simulation that the C.L. of the original LEP sample is only 3.8 \cdot 10^{-4}. We suggest that a reliable extimate of the true R-value is 20.60< R < 20.98 which produces only a very poor determination of the strong coupling constant at the Z mass scale, 0.10< \alpha_s(M_z)< 0.15.Comment: 4 pages, LaTeX; Postscript file, compressed and uuencoded, availabl

Conformal blocks related to the R-R states in the \hat c =1 SCFT

We derive an explicit form of a family of four-point Neveu-Schwarz blocks with $\hat c =1,$ external weights $\Delta_i = 1/8$ and arbitrary intermediate weight. The derivation is based on a set of identities obeyed in the free superscalar theory by correlation functions of fields satisfying Ramond condition with respect to the bosonic (dimension 1) and the fermionic (dimension 1/2) currents.Comment: 15 pages, no figure