The paper addresses the problem of estimating the graph structure of a binary Ising Markov random field using $\ell_1$-regularized logistic regression. The authors propose a method where the neighborhood of each node is estimated by performing logistic regression with an $\ell_1$-constraint. They analyze the method under high-dimensional scaling, where both the number of nodes $p$ and the maximum neighborhood size $d$ grow with the number of observations $n$. The main results provide sufficient conditions on the triple $(n, p, d)$ and the model parameters for consistent neighborhood estimation. Under mild assumptions on the population Fisher information matrix, consistent neighborhood selection is achieved with sample sizes $n = \Omega(d^3 \log p)$ and computational complexity $\mathcal{O}(\max\{n, p\} p^3)$. When these conditions are imposed directly on the sample matrices, a reduced sample size of $n = \Omega(d^2 \log p)$ suffices. The paper focuses on binary Ising models but indicates a generalization to more general discrete Markov random fields.The paper addresses the problem of estimating the graph structure of a binary Ising Markov random field using $\ell_1$-regularized logistic regression. The authors propose a method where the neighborhood of each node is estimated by performing logistic regression with an $\ell_1$-constraint. They analyze the method under high-dimensional scaling, where both the number of nodes $p$ and the maximum neighborhood size $d$ grow with the number of observations $n$. The main results provide sufficient conditions on the triple $(n, p, d)$ and the model parameters for consistent neighborhood estimation. Under mild assumptions on the population Fisher information matrix, consistent neighborhood selection is achieved with sample sizes $n = \Omega(d^3 \log p)$ and computational complexity $\mathcal{O}(\max\{n, p\} p^3)$. When these conditions are imposed directly on the sample matrices, a reduced sample size of $n = \Omega(d^2 \log p)$ suffices. The paper focuses on binary Ising models but indicates a generalization to more general discrete Markov random fields.