Abstract Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For $k$ colours and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-\varepsilon }$ . Following a sequence of gradual improvements for $k\geq n$ colours, the central open question is to resolve the gap between $\Omega (n)$ and …

Since the 1960s Mastermind has been studied for the combinatorial and information theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colors. For $k$ colors and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-ε}$. Following a sequence of gradual improvements for $k \geq n$ colors, the central open question is to resolve the gap between $Ω(n)$ and $\mathcal{O}(n\log \log n)$ f…

We design a randomized algorithm that finds a Hamilton cycle in 𝒪(n) time with high probability in a random graph G_{n,p} with edge probability p ≥ C log n / n. This closes a gap left open in a seminal paper by Angluin and Valiant from 1979.

The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some pro…

We design a randomized algorithm that finds a Hamilton cycle in $\mathcal{O}(n)$ time with high probability in a random graph $G_{n,p}$ with edge probability $p\ge C \log n / n$. This closes a gap left open in a seminal paper by Angluin and Valiant from 1979.