Orthogonal Frequency-Division Multiplexing with Random multiple access (OFDRMA) is discussed for uplink communications, whereby several active users send information towards a single base-station (BS), while all other users are dormant. Originally, uplink communication methods included sharing the frequency resources among the active users in an orthogonal fashion, i.e., a central unit is required to dynamically allocate the resources. More recently, non-orthogonal methods have arisen, meaning that several active users share the same frequency bins, but they still do require a central unit to dynamically allocate the resources in a uniform (as possible) manner over the available bandwidth. The task and overhead required for managing the frequency allocations among the users can be quite cumbersome. In OFDRMA, the frequency allocations for any user are independent of the frequency allocations for the other users, and independent of which of the other users are currently active. Rather, OFDRMA relies on random, yet predetermined, allocation of frequency bins for each user, known only to that user and the BS. A multi-user detection approach is presented based on a graphical representation of the system. It is shown to provide robustness against the forced randomness of the scheme. Capacity of OFDRMA and its optimization are analyzed and provided in detail. Simulation results are provided for demonstrating the performance attainable with OFDRMA and the proposed detection scheme. Both the capacity and the simulations are compared with modern multi-user multiple-input multiple-output (MU-MIMO) schemes.