**How Often Must Curves
Touch?**

Suppose that we are given *n* simple
closed curves in the plane such that each curve shares a point with every other
curve, but no three curves share a common point. Two curves that share a point
need not cross transversally—tangential intersections are allowed. Let *f(**n)* denote the minimum number of
intersections over all such families of curves.

It is obvious that *f(**n)* is at least as large as the binomial
coefficient *C(n,2)*. It is also easy
to construct examples that show *f(n)*
is at most *2 C(n,2)*: take circles *C_i* of the same radius with centers
perturbed a small distance *epsilon_i*
from the origin. It is also known that *f(**4) = 6, f(5) = 12,*
and *f(6) = 20*.

**Problem:** *Determine f(7).*

Asymptotically:

**Conjecture: **Lim f(n)/C(n,2) = 2* (the
limit is as n approaches infinity)*

It is not hard to show that *f(**n+1)* is at least
*ceiling[f(n) (n+1)/(n-1)]*: in any
realization of *n+1* curves, count the
number of induced intersections over all sub-realizations of *n* curves. In combination with *f(**6) = 20*, this implies that the limit in
the conjecture is at least 3/2.

I believe that these problems were first posed by Bruce Richter and Carsten
Thomassen in [RT]. A partial result is given by Gelasio Salazar [S] who showed
that if any two such curves intersect in at most *k* points (for a fixed value of *k*),
then the above limit is 2. This includes many special cases; for example, when
each curve is an ellipse, or when each curve is a piecewise-linear *p*-gon for fixed *p*.

**References: **

[RT] R.B. Richter and C. Thomassen, Intersections of curve systems and the
crossing number of C_5 x C_5, *Disc. Comp.
Geom.* **13** (1995) 149-159.

[S] G. Salazar, On the intersections of systems of
curves, *J. of Combin. Theory Ser. B* **75** (1999) 56-60.

Submitted by: Dan Archdeacon, Dept. of Math. and Stat.,

Send comments to *dan.archdeacon@uvm.edu*

December 2003