We study the following combinatorial counting and sampling problems: can we sample from the Erdős-Rényi random graph$G(n,p)$conditioned on triangle-freeness? Can we approximate (either algorithmically or with a formula) the probability that$G(n,p)$is triangle-free? These are prototypical instances of forbidden substructure problems ubiquitous in combinatorics. The algorithmic questions are instances of approximate sampling and counting for a hypergraph hard-core model. Estimating the probability that$G(n,p)$has no triangles is a fundamental question in probabilistic combinatorics and one that has led to the development of many important tools in the field. Through the work of several authors, the asymnpotics of the logarithm of this probability are known if$p=o(n^{-1/2})$or if$p=\omega(n^{-1/2})$. The regime$p=\Theta(n^{-1/2})$is more mysterious, as this range witnesses a dramatic change in the the typical structural properties of$G(n,p)$conditioned on triangle-freeness. As we show, this change in structure has a profound impact on the performance of sampling algorithms. We give two different efficient sampling algorithms for this problem (and complementary approximate counting algorithms), one that is efficient when$p < c/\sqrt{n}$and one that is efficient when$p > C/\sqrt{n}$for constants$c, C > 0$. The latter algorithm involves a new approach for dealing with large defects in the setting of sampling from low-temperature spin models. Our algorithmic results can be used to give an asymptotic formula for the logarithm of the probability$G(n,p)$is triangle-free when$p < c/\sqrt{n}$. This algorithmic approach to large deviation problems in random graphs is very different than the known approaches in the suBCRitical regime$p=o(n^{-1/2})$(based on the Poisson paradigm) and in the supercritical regime$p=\omega(n^{-1/2})$(based on regularity lemmas or hypergraph containers); in fact, to the best of our knowledge, no asymptotic formula for the log probability in the regime$p=\Theta(n^{-1/2})$was even conjectured previously.

The n-queens puzzle is to place n mutually non-attacking queens on an n × n chessboard. We present a simple two stage randomized algorithm to construct such configurations. In the first stage, a random greedy algorithm constructs an approximate toroidal n-queens configuration. In this well-known variant the diagonals wrap around the board from left to right and from top to bottom. We show that with high probability this algorithm succeeds in placing (1–o(1))n queens on the board. In the second stage, the method of absorbers is used to obtain a complete solution to the non-toroidal problem. By counting the number of choices available at each step of the random greedy algorithm we conclude that there are more than ((1–o(1)) ne–3)n solutions to the n-queens problem. This proves a conjecture of Rivin, Vardi, and Zimmerman in a strong form. Recently, using different methods, Bowtell and Keevash proved the same lower bound for the toroidal problem, giving an independent proof of the result.