Abstract. We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets, B h [g]-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "A contains a non-trivial solution of M ·x = 0" where A is a random set and each of its elements is chosen independently with the same probability from the interval of integers {1, . . . , n}. Our study contains a formal definition of trivial solutions for any linear system, extending a previous definition by Ruzsa when dealing with a single equation.Furthermore, we study the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.

Biased Maker‐Breaker games, introduced by Chvátal and Erdős, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and Łuczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.

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We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Szémeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, Rödl, Ruciński and Schacht based on a hypergraph container approach due to Nenadov and Steger. Lastly we further extend these results to include some solutions with repeated entries using a notion of non-trivial solutions due to Rúzsa as well as Rué et al.

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