On Defining SPARQL with Boolean Tensor Algebra

The Resource Description Framework (RDF) represents information as subject-predicate-object triples. These triples are commonly interpreted as a directed labelled graph. We propose an alternative approach, interpreting the data as a 3-way Boolean tensor. We show how SPARQL queries - the standard queries for RDF - can be expressed as elementary operations in Boolean algebra, giving us a complete re-interpretation of RDF and SPARQL. We show how the Boolean tensor interpretation allows for new optimizations and analyses of the complexity of SPARQL queries. For example, estimating the size of the results for different join queries becomes much simpler

The existence of designs via iterative absorption: hypergraph FF-designs for arbitrary FF

We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs~$F$. This implies that given any $r$-uniform hypergraph~$F$, the trivially necessary divisibility conditions are sufficient to guarantee a decomposition of any sufficiently large complete $r$-uniform hypergraph into edge-disjoint copies of~$F$, which answers a question asked e.g.~by Keevash. The graph case $r=2$ was proved by Wilson in 1975 and forms one of the cornerstones of design theory. The case when~$F$ is complete corresponds to the existence of block designs, a problem going back to the 19th century, which was recently settled by Keevash. In particular, our argument provides a new proof of the existence of block designs, based on iterative absorption (which employs purely probabilistic and combinatorial methods). Our main result concerns decompositions of hypergraphs whose clique distribution fulfills certain regularity constraints. Our argument allows us to employ a `regularity boosting' process which frequently enables us to satisfy these constraints even if the clique distribution of the original hypergraph does not satisfy them. This enables us to go significantly beyond the setting of quasirandom hypergraphs considered by Keevash. In particular, we obtain a resilience version and a decomposition result for hypergraphs of large minimum degree.Comment: This version combines the two manuscripts `The existence of designs via iterative absorption' (arXiv:1611.06827v1) and the subsequent `Hypergraph F-designs for arbitrary F' (arXiv:1706.01800) into a single paper, which will appear in the Memoirs of the AM

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.Comment: 28 page

Rigidity of Frameworks Supported on Surfaces

A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in \bR^2. A more general theory is developed for frameworks in \bR^3 whose vertices are constrained to move on a two-dimensional smooth submanifold \M. Furthermore, when \M is a union of concentric spheres, or a union of parallel planes or a union of concentric cylinders, necessary and sufficient combinatorial conditions are obtained for the minimal rigidity of generic frameworks.Comment: Final version, 28 pages, with new figure