This paper presents results on the typical number of simultaneous point-to-point transmissions above a minimum rate that can be sustained in a network with $n$ transmitter-receiver node pairs when all transmitting nodes can potentially interfere with all receivers. In particular we obtain a scaling law when the fading gains are independent Rayleigh distributed random variables and the transmitters over different realizations are located at the points of a stationary Poisson field in the plane. We show that asymptotically with probability approaching 1, the number of simultaneous transmissions (links that can transmit at greater than a minimum rate) is of the order of $O(n^{\frac{1}{4}})$. These asymptotic results are confirmed from simulations.