Abstract Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions.

Abstract The thin set theorem for n-tuples and k colors ( $\operatorname {\mathrm {\sf {TS}}}^n_k$ ) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$ , nor the free set theorem ( $\operatorname {\mathrm {\sf {FS}}}^n$ ) imply the Erdős–Moser theorem ( $\operatorname {\mathrm {\sf {EM}}}$ ) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$ , a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$ -instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).