A crossing family in a finite planar point set 
in general position, with
no three collinear points, is a collection of line
segments with endpoints in 
whose every pair intersects away from the endpoints of both
segments. Shared endpoints do not count as crossings (Aronov et al. 1994).
Orthaber and Schnider (2026) proved that every such 
-point set contains a crossing family of size at least
 |
where 
is the floor function. This gives an explicit universal
linear lower bound, with asymptotic coefficient 
. The geometric notion differs from crossing
families of sets in combinatorial optimization.
Let 
be the largest crossing-family size and 
the number of crossings in the complete straight-line drawing
on 
.
Orthaber and Schnider (2026) also proved, for 
,
 |
The count 
belongs to the fixed drawing, rather than being the minimum
over all drawings in the rectilinear crossing
number.
