We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$. For $r = 2$ this improves a bound due to Alon and is tight. For $r \ge 3$, previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows that, for each $\ell$-uniform hypergraph $H$ of order $k$, there is a collection $C$ of $\ell$-uniform hypergraphs of order $n$ each with $o(n^k)$ copies of $H$, such that every $H$-free $\ell$-uniform hypergraph of order $n$ is a subgraph of a hypergraph in $C$, and $\log |C| \le c n^{\ell-1/m(H)} \log n$ where $m(H)$ is a standard parameter (there is a similar statement for induced subgraphs). This yields simple proofs, for example, for the number of $H$-free hypergraphs, and for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields a counting version of the K{\L}R conjecture. Likewise, for systems of linear equations the containers supply, for example, bounds on the number of solution-free sets, and the existence of solutions in sparse random subsets. Balogh, Morris and Samotij have independently obtained related results

One of the most basic questions one can ask about a graph H is: how many H-free graphs on n vertices are there? For non-bipartite H, the answer to this question has been well-understood since 1986, when Erdős, Frankl and Rödl proved that there are 2 (1+o(1))ex(n,H) such graphs. For bipartite graphs, however, much less is known: even the weaker bound 2 O(ex(n,H)) has been proven in only a few special cases: for cycles of length four and six, and for some complete bipartite graphs.For even cycles, Bondy and Simonovits proved in the 1970s that ex(n, C 2ℓ ) = O n 1+1/ℓ , and this bound is conjectured to be sharp up to the implicit constant. In this paper we prove that the number of C 2ℓ -free graphs on n vertices is at most 2 O(n 1+1/ℓ ) , confirming a conjecture of Erdős. Our proof uses the hypergraph container method, which was developed recently (and independently) by Balogh, Morris and Samotij, and by Saxton and Thomason, together with a new 'balanced supersaturation theorem' for even cycles. We moreover show that there are at least 2 (1+c)ex(n,C6) C 6 -free graphs with n vertices for some c > 0 and infinitely many values of n ∈ N, disproving a well-known and natural conjecture. As a further application of our method, we essentially resolve the so-called Turán problem on the Erdős-Rényi random graph G(n, p) for both even cycles and complete bipartite graphs.Research supported in part by a CNPq bolsa PDJ (DS) and by CNPq Proc. 479032/2012-2 and Proc. 303275/2013-8 (RM). 1 1.1. History and background. The study of extremal graph theory was initiated roughly 70 years ago by Turán [52], who determined exactly the extremal number of the complete graph, by Erdős and Stone [29], who determined asymptotically (for all r 3) the extremal number of a complete r-partite graph 1 , and by Kővári, Sós and Turán [41] who showed that ex(n, K s,t ) = O(n 2−1/s ), where K s,t denotes the complete bipartite graph with part sizes s and t. (The case K 2,2 = C 4 was solved some years earlier by Erdős [21] during his study of multiplicative Sidon sets.) Over the following decades, a huge amount of effort was put into determining more precise bounds for specific families of graphs (see, e.g., [11,31]), and a great deal of progress has been made. Nevertheless, the order of magnitude of ex(n, H) for most bipartite graphs, including simple examples such as K 4,4 and C 8 , remains unknown.In the 1970s, the problem of determining the number of H-free graphs on n vertices was introduced by Erdős, Kleitman and Rothschild [23], who proved that there are 2 (1+o(1))ex(n,Kr) K r -free graphs, and moreover that almost all triangle-free graphs are bipartite. This latter result was extended to all cliques by Kolaitis, Prömel and Rothschild [39] and to more general graphs by Prömel and Steger [46], and the former to all non-bipartite graphs by Erdős, Frankl and Rödl [24], using Szemerédi's regularity lemma. The corresponding result for k-uniform hypergraphs was proved by Nagle, Rödl and Schacht [44] using hypergraph regularity, and reproved by Balogh,...

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Mathematical reasoning-a core ability within human intelligence-presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis of inferring, learning, and exploiting laws, axioms, and symbol manipulation rules. In this paper, we present a new challenge for the evaluation (and eventually the design) of neural architectures and similar system, developing a task suite of mathematics problems involving sequential questions and answers in a free-form textual input/output format. The structured nature of the mathematics domain, covering arithmetic, algebra, probability and calculus, enables the construction of training and test splits designed to clearly illuminate the capabilities and failure-modes of different architectures, as well as evaluate their ability to compose and relate knowledge and learned processes. Having described the data generation process and its potential future expansions, we conduct a comprehensive analysis of models from two broad classes of the most powerful sequence-to-sequence architectures and find notable differences in their ability to resolve mathematical problems and generalize their knowledge.