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We investigate packet-by-packet rate adaptation so as to maximize the throughput. We consider a finite-state Markov channel (FSMC) with collisions, which models channel fading as well as collisions due to multi-user interference. To limit the amount of feedback data, we only use past packet acknowledgements (ACKs) and past rates as channel state information. The maximum achievable throughput is computationally prohibitive to determine, thus we employ a two-pronged approach. Firstly, we derive new upper bounds on the maximum achievable throughput, which are tighter than previously known ones. Secondly, we propose the particle-filter-based rate adaptation (PRA), which employs a particle filter to estimate the a posteriori channel distribution. The PRA can easily be implemented even when the number of available rates is large. Numerical studies show that the PRA performs within one dB of SNR to the proposed upper bounds for a slowly time-varying channel, even in the presence of multi-user interference.