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.

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Correspondence to J. H. Kim.

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* Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.

Part of this work was done at AT&T Bell Labs and Microsoft Research

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Kim, J.H., Vu*, V.H. Small Complete Arcs in Projective Planes. Combinatorica 23, 311–363 (2003). https://doi.org/10.1007/s00493-003-0024-1

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AMS Subject Classification (2000):

  • 05B25