AbstractWe define and study a new notion calledk‐immunity that lies between immunity and hyperimmunity in strength. Our interest ink‐immunity is justified by the result that θ does notk‐tt reduce to ak‐immune set, which improves a previous result by Kobzev [7]. We apply the result to show that Φ′ does not btt‐reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of effectively simple sets and show that simple sets are not