# 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.

* 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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• 05B25