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We present a new inner bound for the admissible rate region of the t-stage successive-refinement problem with side-information. We also present a new upper bound for the rate-distortion function for lossy-source coding with multiple receivers and side-information. A single-letter characterisation of this rate-distortion function is a long-standing open problem, and it is widely believed that the tightest upper bound is provided by Theorem 2 of Heegard and Berger's paper &#x201C;Rate Distortion when Side Information may be Absent,&#x201D; IEEE Trans. Inform. Theory, 1985. We give a counterexample to Heegard and Berger's result.

Rate distortion with Side-Information at many receivers