Independent sets in hypergraphs and Ramsey properties of graphs and the integers

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris, and Samotij [J. Amer. Math. Soc., 28 (2015), pp. 669--709], and independently Saxton and Thomason [Invent. Math., 201 (2015), pp. 925--992], developed very general container theorems for independent sets in hypergraphs, both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular we do the following: (a) We generalize the random Ramsey theorem of Rödl and Ruciński [Combinatorics, Paul Erdös Is Eighty, Vol. 1, Bolyai Soc. Math. Stud., János Bolyai Mathematical Society, Budapest, 1993, pp. 317--346; Random Structures Algorithms, 5 (1994), pp. 253--270; J. Amer. Math. Soc., 8 (1995), pp. 917--942] by providing a resilience analogue. Our result unifies and generalizes several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers [Ann. of Math., 184 (2016), pp. 367--454] and Schacht [Ann. of Math., 184 (2016), pp. 331--363]. (b) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter [Random Structures Algorithms, 11 (1997), pp. 245--276]. (c) All of the above results in fact hold for uniform hypergraphs. (d) For a (hyper)graph $H$, we determine, up to an error term in the exponent, the number of $n$-vertex (hyper)graphs $G$ that have the Ramsey property with respect to $H$ (that is, whenever $G$ is $r$-colored, there is a monochromatic copy of $H$ in $G$). (e) We strengthen the random Rado theorem of Friedgut, Rödl, and Schacht [Random Structures Algorithms, 37 (2010), pp. 407--436] by proving a resilience version of the result. (f) For partition regular matrices $A$ we determine, up to an error term in the exponent, the number of subsets of $\{1,\dots,n\}$ for which there exists an $r$-coloring which contains no monochromatic solutions to $Ax=0$. Along the way a number of open problems are posed

A Note on sparse supersaturation and extremal results for linear homogeneous systems

We study the thresholds for the property of containing a solution to a linear homogeneous system in random sets. We expand a previous sparse Sz\'emeredi-type result of Schacht to the broadest class of matrices possible. We also provide a shorter proof of a sparse Rado result of Friedgut, R\Postprint (published version

On the optimality of the uniform random strategy

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.Comment: 26 page