ComPoSe — Combinatorics of Point Sets and Arrangements of Objects
This CRP focuses on combinatorial properties of discrete sets of
points and other simple geometric objects primarily in the plane. In
general, geometric graphs are a central topic in discrete and
computational geometry, and many important questions in mathematics
and computer science can be formulated as problems on geometric
graphs. In the current context, several families of geometric graphs,
such as proximity and skeletal structures, constitute useful
abstractions for the study of combinatorial properties of the point
sets on which they are defined. For arrangements of other objects,
such as lines or convex sets, their combinatorial properties are
usually also described via an underlying graph structure.
The following four tasks are well-known hard problems in this area and
will form the backbone of the current project. We will consider the
intriguing class of Erdős-Szekeres type problems, variants of
graph problems with colored vertices, counting and enumeration
problems for specific classes of geometric graphs, and generalizations
of order types as a versatile tool to investigate the combinatorics of
point sets. All these problems are combinatorial problems on geometric
graphs and are interrelated in the sense that approaches developed for
one of them will also be useful for the others. Moreover, progress in
one direction might provide a better understanding for related
questions. Our main objective is to gain deeper insight into the
structure of this type of problems and to contribute major steps
towards their final solution.
Erdős-Szekeres problems. We will investigate specific
variants of this famous group of problems, such as colored versions,
and use newly developed techniques, such as a recent generalized
notion of convexity, to progress on this topic. A typical example is
the convex monochromatic quadrilateral problem in Section (iv) of the
Call for Outline Proposals: Prove or disprove that every (sufficiently
large) bichromatic point set contains an empty convex monochromatic
quadrilateral. Recent progress on this and other Erdős-Szekeres
type problems has been made by the PIs Aichholzer, Hurtado, Pach,
Valtr, and Welzl.
Colored point sets. An interesting family of questions is the
existence of constrained colorings of point sets. We may consider, for
instance, the problem of coloring a set of points in a way such that
any unit disk with sufficiently many points contains all colors. Also,
colored versions of classical Helly-type results continue to be a
source of fundamental problems, requiring the use of combinatorial and
topological tools. In particular we are interested in colored
versions of Tverberg-type results and their generalization of
Tverberg-Vrećica-type. Pach founded the class of ‘covering
colored sets’ problems and will cooperate on these problems with
Cardinal and Felsner in particular, but also with all other PIs.
Counting, enumerating, and sampling of crossing-free
configurations. Planar graphs are a core topic in abstract graph
theory. Their counterpart in geometric graph theory are crossing-free
(plane) graphs. Interesting questions arise from considering specific
classes of plane graphs, such as triangulations, spanning cycles,
spanning trees, and matchings. For example, the flip-graph of the set
of all graphs of a given class allows a fast enumeration of all
elements from this class and even efficient optimization with respect
to certain criteria. But when it comes to more intricate demands, like
counting or sampling a random element, very little is understood. We
will put emphasis on counting, enumerating, and sampling methods for
several of the mentioned graph classes. Related extremal results
(e.g. upper bounds on the number of triangulations) will also be
considered for other classes, like string graphs of a fixed order k
(intersection graphs of curves in the plane with at most k
intersections per pair) or visibility graphs in the presence of at
most k connected obstacles. Aichholzer, Hurtado, and Welzl have been
involved in recent progress on lower and upper bounds for the number
of several mentioned classes of geometric graphs and will cooperate
with Pach (intersection graphs), Valtr, and Felsner (higher
dimensions) on enumerating and counting.
Order types (rank 3 oriented matroids). Order types play a
central role in the above mentioned problems, and constitute a useful
tool to investigate the combinatorics of point sets. This is done,
e.g., by providing small instances of vertex sets for extremal
geometric graphs in enumeration problems. Our goal is to generalize,
and at the same time specialize, this concept. For example, we plan to
investigate the k-set problem as well as a generalization of the
Erdős-Szekeres theorem for families of convex bodies in the
plane. Typically, progress on the k-set problem has frequently been
achieved in the language of pseudoline arrangements, which are dual to
order types. In particular we are interested in combinatorial results
ranging from Sylvester-type results to counting certain cells, and the
number and structure of arrangements of n pseudo-lines. Felsner is an
expert on pseudo-line arrangements and will collaborate here with
Valtr, Pach, Welzl and Aichholzer on order types. Moreover all PIs
have been working on the kset problem individually and will make a
joint afford.
This CRP tackles fundamental questions at the intersection of
mathematics and theoretical computer science. It is well known that in
this area some problems require only days to be solved, others may
take decades or even more. Thus, the working schedule with respect to
obtaining the desired theoretical results must follow the standard
approach: Continuation of work in progress, evaluation of the results
obtained by other authors and groups, and continuous identification of
new directions for progress and exploration, hence always advancing
the frontiers of knowledge. Since it is infeasible to impose a proper
temporal order on the objectives and milestones to be attained - the
conceptual implications are manifold, and many of the stated
objectives are strongly interrelated - it will be the very progress of
research and the obtained results that mark our progress in time. This
is guaranteed by the competence of the team. The major 'visible'
milestones will be the regular presentations of joint papers in the
main conferences of the field, the corresponding submissions to
journals, and a series of progress reports that will help in keeping a
clear and consistent guidance and interaction with the other teams.
Several of the mentioned problems are long-standing open questions and
known to be hard. Therefore we will consider several specific variants
of them to determine how far state-of-the-art methods can be used and
where new approaches have to be found. This will definitely improve
our understanding of the structure of these problems, with the goal of
making major contributions towards their solution or, in the ideal
case, to finally settle them. Most of our approaches will be of
theoretical nature. But we will also make intensive use of computers
for enumeration and experiments, to get initial insights into the
structure of problems, or to support or refute conjectures.
It is well known that the mentioned problems have resisted several
previous attacks and therefore require the cooperation of researchers
with strong and complementary expertise. We consider large-scale
collaboration on these topics as one of the main ingredients for
success. Thus we will not have individual projects running in
parallel, but all participants will jointly work on the topics, in a
massive collaborative effort. To guarantee a strong interaction
between the members of the group we will maintain regular exchanges of
senior researchers and students, regular joint research workshops (1-2
per year), and frequent visits.