In 1958, Hill conjectured that the minimum number of crossings in a drawing of K_n K n is
exactly Z (n)= 1 4\big ⌊ n 2\big ⌋\big ⌊ n-1 2\big ⌋\big ⌊ n-2 2\big ⌋\big ⌊ n-3 2\big ⌋ Z (n) …

The calculation of the rectilinear crossing number of complete graphs is an important open
problem in combinatorial geometry, with important and fruitful connections to other classical …

Let P be a set of points in general position in the plane. Join all pairs of points in P with
straight line segments. The number of segment-crossings in such a drawing, denoted by …

62] Turan, P., A note of welcome, J. Graph Theory 1 (1977) 7-9. 63] Dambitis, J., An
algorithm for superimposing a nonplanar graph onto the plane with nearly minimal number …

Let $\crs (K_n) $ be the minimum number of crossings over all rectilinear drawings of the
complete graph on $ n $ vertices on the plane. In this paper we prove that $\crs (K_n)< …

Even the most superficial glance at the vast majority of crossing-minimal geometric drawings
of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold …

For sets of n points, n even, in general position in the plane, we consider straight-line
drawings of perfect matchings on them. It is well known that such sets admit at least C n/2 …

We apply ideas from the theory of limits of dense combinatorial structures to study order
types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain …

We consider a variation of the classical Erdős–Szekeres problems on the existence and
number of convex k-gons and k-holes (empty k-gons) in a set of n points in the plane …

A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are
drawn as straight-line segments. The rectilinear crossing number of a graph is the minimum …