• Corpus ID: 248157303

# The Dirac-Goodman-Pollack Conjecture

@inproceedings{Dumitrescu2022TheDC,
title={The Dirac-Goodman-Pollack Conjecture},
year={2022}
}
In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the…

## References

SHOWING 1-10 OF 69 REFERENCES
Three point collinearity
• American Mathematical Monthly
• 1943
Multiple intersections of diagonals of regular polygons, and related topics
Our main concern is to investigate geometrically all sets of three concurrent chords of regular polygons or, equivalently, all adventitious quadrangles (that is, all quadrangles such that the angle
On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry
It will be shown here that for some point of S the number of connecting lines through it exceedsc · n, and the following conjecture of Erdős is proved: If any straight line contains at mostn−x points of S, then the numberof connecting lines determined byS is greater thanc · x · n.
Extremal problems in discrete geometry
• Mathematics
Comb.
• 1983
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
Crossing Numbers and Hard Erdős Problems in Discrete Geometry
• Mathematics
• 1997
We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the
A Pseudoline Counterexample to the Strong Dirac Conjecture
• Mathematics
Electron. J. Comb.
• 2014
We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. This shows the "Strong Dirac"
A combinatorial perspective on some problems in geometry
• Congressus Numerantium
• 1981
New Lower Bounds for the Number of Pseudoline Arrangements
• Computer Science
SODA
• 2019
It is shown that b_n \geq cn^2 -O(n \log{n})$for some constant$c>0.2083$for large$n, which improves the previous best lower bound due to Felsner and Valtr (2011).
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
• Mathematics, Computer Science
Discret. Comput. Geom.
• 2006
If a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5( v-2); and the crossing number of any graph is at least $\frac73e-\frac{25}3(v-2).$ Both bounds are tight up to an additive constant.