### Description

### 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

configurations.

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.

Status | Finished |
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Effective start/end date | 1/10/11 → 31/12/15 |