# Why are there no maximal arcs of odd order?

A projective plane is a point/line incidence geometry that satisfies the following conditions:

• Every two points lie on a unique line
• Every two lines meet in a unique point
• There is a quadrangle (four points, no three of which are collinear)

The last condition is just a non-degeneracy condition to eliminate trivial cases such as a single line containing all the points, and I’ll also add that I am only concerned here with finite projective planes. In this case it is easy to show that there is a number $s$ such that there are $s^2+s+1$ points, $s^2+s+1$ lines, each line contains $s+1$ points, each point lies in $s+1$ lines; this special number $s$ is called the order of the plane. In design theory language a projective plane is just a $2-(s^2+s+1,s+1,1)$ design.

The “canonical example” of a projective plane is the Desarguesian plane $PG(2,q)$ whose point set is the set of 1-dimensional subspaces of the vector space $V = GF(q)^3$ and line set is the set of 2-dimensional subspaces of $V$. The order of $PG(2,q)$ is $q$ and since there is a field $GF(q)$ for every prime power $q$, there is a projective plane of every prime power order. Probably the main open question in finite geometry is

Is there a finite projective plane whose order is not a prime power?

One thing that makes this problem difficult is that there are many many projective planes known other than $PG(2,q)$ – lots of infinite families and plenty of  examples of small order – but they all resolutely have prime power order. (For those interested, Eric Moorhouse keeps comprehensive lists of small order planes.) If $PG(2,q)$ were the only projective plane that was known, then we would conclude that the combinatorial definition of a projective plane was somehow so restrictive that it could only be met by the algebraic structure of a field. However, many of the known projective planes appear to have nothing much to do with fields and so the whole question is more subtle than just the existence of a field – in particular, what is it about projective planes that forces the orders of the fields to occur but dispenses with the fields themselves! The observant reader will notice that I am writing as though it is obvious that the answer to the question is “No” but, while this is what I personally believe, there are many geometers much more talented than I who believe the opposite, and spend at least some “Friday afternoon time” looking for planes of order 12 or 20 or 56 or whatever.

### Maximal Arcs

Much more is known about the orders for which projective planes may or may not exist, but this is all written down elsewhere and today I want to discuss a less well-known problem, which is the existence of maximal arcs in projective planes. A maximal arc is a subset $\mathcal{K}$ of the points of a projective plane such that every line that meets $\mathcal{K}$ does so in the same number of points. In other words, there is a number $n$ such that every line of the plane contains either $0$ of $n$ points of $\mathcal{K}$ – we call this a set of type $(0,n)$. Some simple counting shows that if a plane of order $s$ contains a set of type $(0,n)$ then $n$ must divide $s$. [Edit: as JB points out, we should insist that $n$ be strictly less than $s$.]

So when does these sets of points actually occur? For planes of even order the answer is pretty simple – the Desarguesian plane $PG(2,2^h)$ contains a set of type $(0,n)$ for every possible value of $n$ that satisfies the necessary condition. Computer searches on the planes of order $16$ show that most of them contain sets of type $(0,n)$.

For planes of odd order, the situation is very different – not a single maximal arc in any projective plane of odd order is known. Often if there is an even-odd dichotomy of this sort, there is a parity argument hiding somewhere, but in this case nobody has managed to find one. However it is known that the Desarguesian plane $PG(2,q)$ with $q$ odd does not contain any maximal arcs. This was proved by Ball, Blokhuis and Mazzoca (paper here) who gave a very algebraic proof based on properties of the field underlying $PG(2,q)$.

But what about the other planes? If it really is the case that maximal arcs do not exist in any projective plane of odd order, then this is a combinatorial statement for which we would really like a combinatorial proof. On the other hand, if it is only the algebra of $PG(2,q)$ that prevents a maximal arc appearing there, then surely we should be able to find an example in some of the hundreds of thousands of known non-Desarguesian planes!

So this leads us to the main problem:

Find a maximal arc in a non-Desarguesian plane of odd order, or give a combinatorial reason why they cannot exist.

## 2 thoughts on “Why are there no maximal arcs of odd order?”

1. John Bamberg says:

John Sheekey pointed out to me that we should have $n , otherwise we get trivial examples by removing a line from $PG(2,q)$ to get $(0,q)$-sets.