The geometry of hyperbolic lines in polar spaces

In this paper we consider partial linear spaces induced on the point set of a polar space, but with as lines the hyperbolic lines of this polar space. We give some geometric characterizations of these and related spaces. The results have applications in group theory, in the theory of Lie algebras and in graph theory

Bidimensionality of Geometric Intersection Graphs

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes

Rank of divisors on graphs: an algebro-geometric analysis

The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in some cases.Comment: Final version incorporating referee remarks. Dedicated to Joe Harris for his sixtiest birthday. 23 page

Regularity lemmas in a Banach space setting

Szemer\'edi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lov\'asz and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst, Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compact- ness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lov\'asz and Szegedy as well as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions, arXiv preprint arXiv:1401.2906 (2014)].Comment: 15 pages. The topological part has been substantially improved based on referees comments. To appear in European Journal of Combinatoric