Small Complete Arcs in Projective Planes

In the late 1950’s, B. Segre introduced the fundamental notion of arcs and complete arcs [48,49]. An arc in a finite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane ${\user1{P}}$, determining $n{\left( {\user1{P}} \right)}$, the size of its smallest complete arc, has been a major open question in finite geometry for several decades. Assume that ${\user1{P}}$ has order q, it was shown by Lunelli and Sce [41], more than 40 years ago, that ${\left( {\user1{P}} \right)} \geqslant {\sqrt {2q} }$. Apart from this bound, practically nothing was known about $n{\left( {\user1{P}} \right)}$ , except for the case ${\user1{P}}$ is the Galois plane. For this case, the best upper bound, prior to this paper, was O(q ^3/4) obtained by Szőnyi using the properties of the Galois field GF(q).

In this paper, we prove that $n{\left( {\user1{P}} \right)} \leqslant {\sqrt q }{\kern 1pt} \log ^{c} q$ for any projective plane ${\user1{P}}$ of order q, where c is a universal constant. Together with Lunelli-Sce’s lower bound, our result determines $n{\left( {\user1{P}} \right)}$ up to a polylogarithmic factor. Our proof uses a probabilistic method known as the dynamic random construction or Rödl’s nibble. The proof also gives a quick randomized algorithm which produces a small complete arc with high probability.

The key ingredient of our proof is a new concentration result, which applies for non-Lipschitz functions and is of independent interest.