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Cooperative transmission among wireless network nodes can be exploited to resolve collisions and thereby enhance the network throughput. Incorporation of multi-beam adaptive array (MBAA) at a base station/access point (destination) receiver has been shown to improve the network performance. In this paper, we propose an efficient cooperative wireless medium access scheme that exploits novel relay selection methods in a network equipped with MBAA at the destination receiver. Unlike existing techniques that require the estimation of angles-of-arrival (AoAs), the proposed scheme uses the spatial correlation among users for simpler but more effective collision detection and resolution. We present two useful relay selection methods based on channel gain and spatial correlation. It is shown that the joint use of an effective relay selection method and an MBAA in a wireless network can significantly improve the uplink throughput. The throughput of the proposed scheme and its upper bound are analytically derived. Numerical and simulation results have demonstrated significant performance enhancement achieved by the proposed cooperative wireless medium access scheme.