# Geometry and Combinatorics

## Complex and hypercomplex Sylvester-Gallai theorems

The following is more-or-less the content of a talk I gave at the Sächsischer Geometrietag on 7 December 2007 at the University of Magdeburg.

In a previous post I discussed the Sylvester-Gallai theorem. It says the following:

Theorem 1. Let $\mathcal{S}$ be a finite set of points in $\mathbb{R}^n$. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Then $\mathcal{S}$ spans an affine subspace of dimension $1$.

In that post I suggested that Sylvester was motivated by a 2-dimensional counterexample of the 9 inflection points of a nonsingular cubic curve in the complex plane. Probably motivated by the same example, Jean-Paul Serre asked the following in the problem section of the American Mathematical Monthly (1961):

Let $\mathcal{S}$ be a finite set of points in $\mathbb{C}^n$. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Does it follow that $\mathcal{S}$ spans an affine subspace of dimension at most $2$?

No solution to this problem ever appeared in the Monthly. In 1986 L. M. Kelly published a proof of the following theorem.

Theorem 2. Let $\mathcal{S}$ be a finite set of points in $\mathbb{C}^n$. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Then $\mathcal{S}$ spans a space of dimension at most $2$.

Kelly’s proof uses a deep inequality of Hirzebruch (1983) involving line arrangements in the complex projective plane. Around the same time Elkies found an elementary proof that generalizes one of the many proofs of the (real) Sylvester-Gallai theorem. He never published it. In 2002 Lou Pretorius and I asked him for a copy of his preprint. Although the proof was clearly elementary, one of the key lemmas was somewhat daunting, with a proof that was a tour de force. Finally, at the beginning of 2004, instead of indefinitely postponing reading the proof, we decided to give up and try to find our own proof. We did find a very simple and intuitive proof, in fact so simple, that we immediately started working on the following problem.

Let $\mathcal{S}$ be a finite set of points in $\mathbb{H}^n$, the $n$-dimensional vector space over the skew-field of quaternions. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Bound the dimension of the affine subspace spanned by $\mathcal{S}$.

The following was the result, appearing in a joint paper with Elkies, together with his proof for the complex numbers (and for good measure, including a proof for the real case as well!).

Theorem 3. Let $\mathcal{S}$ be a finite set of points in $\mathbb{H}^n$. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$. Then $\mathcal{S}$ spans a space of dimension at most $3$.

Unfortunately, no 3-dimensional example is known, i.e., a finite set of points spanning $\mathbb{H}^3$ such that the line through any two of the points passes through a third. Pretorius and I tried to find lower bounds on the number of points such an example should have, and we came up with the following:

Theorem 4. Let $\mathbb{D}$ be any skew-field. Let $\mathcal{S}$ be a finite set of points in $\mathbb{D}^n$ spanning an affine subspace of dimension at least 3. Suppose that the line through any two points of $\mathcal{S}$ passes through a third point of $\mathcal{S}$.

Then $\mathcal{S}$ has at least 15 points, with equality iff $\mathbb{D}$ has characteristic 2 and $\mathcal{S}$ has the same structure as the projective 3-space over the field with 2 elements.

If the characteristic of $\mathbb{D}$ is not 2, then $\mathcal{S}$ has at least 27 points, with equality iff $\mathbb{D}$ has characteristic 3 and $\mathcal{S}$ has the same structure as the affine 3-space over the field with 3 elements.

If the characteristic of $\mathbb{D}$ is not 2 or 3, then $\mathcal{S}$ has at least 51 points.

Thus a 3-dimensional example over the quaternions would need at least 51 points. We got stuck with a combinatorial explosion, which prevented us going higher. However, the lower bound of 51 cannot be improved beyond 75 without making any extra assumptions on $\mathbb{D}$. as the following example shows: the set consisting of the $3\times 25$ points on 3 parallel affine planes in the 3-dimensional affine space over the field of 5 elements

Another approach is to stay in dimension 2, and then try to find a line passing through some two points of a non-collinear set, but through as few as possible. The Sylvester-Gallai theorem is an example, in the following reformulation:

Theorem 1′. Let $\mathcal{S}\subset\mathbb{R}^2$ be finite and noncollinear. Then there exists a line $\ell$ with $\#(\mathcal{S}\cap\ell)=2$.

In the proof of Theorem 2, Kelly needed only the following consequence of the inequality of Hirzebruch, which gives the analogous complex statement:

Theorem 5. Let $\mathcal{S}\subset\mathbb{C}^2$ be finite and noncollinear. Then there exists a line $\ell$ with $2\leq \#(\mathcal{S}\cap\ell)\leq 3$.

The upper bound 3 here is of course best possible, as already mentioned. A problem that I find fascinating is to find an elementary proof of Theorem 5 (or even more optimistically, of the inequality of Hirzebruch).

Here are two attempts (joint work with József Solymosi). In the first we just try to find some upper bound. In the second we assume that the set $\mathcal{S}$ is a grid, i.e., a Cartesian product of two sets of complex numbers.

Theorem 6.

I. Let $\mathcal{S}\subset\mathbb{C}^2$ be finite and noncollinear. Then there exists a line $\ell$ with $2\leq \#(\mathcal{S}\cap\ell)\leq 5$.

II. Let $\mathcal{S}\subseteq A\times B$ with $A,B\subset\mathbb{C}$ finite. Then there exists a line $\ell$ with $\#(\mathcal{S}\cap\ell)= 2$.

As before, if something can be done easily for complex numbers, then why not try the quaternions? Proofs for the following also appear in my paper with Solymosi.

Theorem 7.

I. Let $\mathcal{S}\subset\mathbb{H}^2$ be finite and noncollinear. Then there exists a line $\ell$ with $2\leq \#(\mathcal{S}\cap\ell)\leq 24$.

II. Let $\mathcal{S}\subseteq A\times B$ with $A,B\subset\mathbb{H}$ finite. Then there exists a line $\ell$ with $2\leq \#(\mathcal{S}\cap\ell)\leq 5$.

Most likely the upper bounds here are much too large.

Bibliography

1. N. Elkies, L. M. Pretorius, K. J. Swanepoel, Sylvester-Gallai Theorems for Complex Numbers and Quaternions, Discrete Comput. Geom. 35 (2006), 361-373.
2. F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II, Birkhäuser Boston, Mass., 1983, pp. 113-140.
3. L. M. Kelly, A resolution of the Sylvester-Gallai problem of J.-P. Serre, Discrete Comput. Geom. 1 (1986), 101-104.
4. J. P. Serre, Problems, Amer. Math. Monthly 73 (1966), 89.
5. L. M. Pretorius and K. J. Swanepoel, The Sylvester-Gallai theorem, colourings and algebra, to appear in Discrete Mathematics.
6. J. Solymosi and K. J. Swanepoel, Elementary incidence theorems for complex numbers and quaternions, submitted.