Small Complete Arcs in Projective
Planes |
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Authors: | Email author" target="_blank">J?H?KimEmail author V?H?Vu |
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Institution: | (1) Microsoft Research, Microsoft Corporation, Redmond, WA 98052, USA;(2) Department of Mathematics, UCSD, La Jolla, CA 92093, USA |
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Abstract: | In the late 1950 s, B. Segre introduced the fundamental
notion of arcs and complete arcs 48,49]. An arc in a nite
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
, determining
, the size of its
smallest complete arc, has been a major open question in finite
geometry for several decades. Assume that
has order
q, it was shown by Lunelli
and Sce 41], more than 40 years ago, that
. Apart from this bound,
practically nothing was known about
, except for the case
is the Galois plane. For
this case, the best upper bound, prior to this paper, was
O(q
3/4)
obtained by Sz nyi using the properties of the Galois field
GF(q).In this paper, we prove that
for any projective plane
of order
q, where
c is a universal constant.
Together with Lunelli-Sce s lower bound, our result determines
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.* 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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Keywords: | 05B25 |
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