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Wireless Mesh Networks (WMNs) represent a new and promising paradigm that uses multi-hop communications to extend WiFi networks: By deploying only one hot spot (directly connected to the Internet) and several transient access points (TAPs), an Internet Service Provider (ISP) can extend its coverage and serve a large number of clients using a single broadband connection. Unfortunately, if the medium access protocol is poorly designed or inadequate, it can lead to severe unfairness and low bandwidth utilization. In this paper, we propose a fair scheduling mechanism that optimizes the bandwidth utilization in the mesh network. Our solution assigns transmission rights to the links in the WMN and maximizes the spatial reuse (i.e., the possibility for links that do not contend to be activated at the same time). We show that our solution is fair and collision-free, and we evaluate its efficiency by means of simulations.