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.

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.