Erdős, Fajtlowicz and Staton asked for the least integer f(k) such that every graph with more than f(k) vertices has an induced regular subgraph with at least k vertices. Here we consider the following relaxed notions. Let g(k) be the least integer such that every graph with more than g(k) vertices has an induced subgraph with at least k repeated degrees and let h(k) be the least integer such that every graph with more than h(k) vertices has an induced subgraph with at least k maximum degree vertices. We obtain polynomial lower bounds for h(k) and g(k) and nontrivial linear upper bounds when the host graph has bounded maximum degree.