- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Geometry

Volume 2013 (2013), Article ID 589362, 3 pages

http://dx.doi.org/10.1155/2013/589362

## Note on a Class of Subsets of AG(3, *q*) with Intersection Numbers 1, *q* and *n* with respect to the Planes

Dipartimento di Ingegneria Civile, Seconda Università degli Studi di Napoli, Real Casa dell’Annunziata, Via Roma 29, 81031 Aversa, Italy

Received 28 June 2013; Accepted 8 October 2013

Academic Editor: Changzheng Qu

Copyright © 2013 Vito Napolitano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give a new and correct proof of a result of O. Ferri and S. Ferri (1995) on -caps of in this paper; moreover we prove that sets of of type with respect to the planes of have size at most with equality if and only if is a cap.

#### 1. Introduction

Let denote either a finite projective space or a finite affine space, and let be a set of nonnegative integers with . A subset of is of class with respect to the subspaces of dimension of if any subspaces intersect either in or points, and is of type with respect to the subspaces of dimension if for every integer , there exists a -subspace meeting in exactly points. The numbers are called *intersection numbers* of . As usual, by a -set we mean a set of size .

In the literature one can find many papers devoted to the study of -sets of given type, not only in affine and projective geometries (cf., e.g., [1–15]), and most of these results are characterizations of classical geometric objects. Recently, characterizations of Hermitian varieties and quadrics of as -sets with given intersection numbers with respect to more than one family of subspaces (e.g., with respect to planes and solids) have been considered [16, 17].

A *cap* of an affine or projective space of dimension ≥3 is a subset of points no three of which are collinear.

In 1995, O. Ferri and S. Ferri [18] gave a characterization of -caps of in terms of sets with three given intersection numbers with respect to the planes. Their result reads as follows.

Theorem 1 (see [18]). *Let be a subset of with points and of type . Then, and is a cap of .*

Unfortunately, a step of the proof of that theorem is not correct; in fact it contains a counting argument which does not give the contradiction they want (see [18] page 71 line +7). However, the statement of the result is true as we are going to prove in Lemma 4.

In this paper we will prove the following slight extension of the O. Ferri and S. Ferri result.

Theorem 2. *Let be a set of of size and with three intersection numbers 1, , and . Then , and if and only if . Moreover, if the set is a cap.*

Thus, it follows that the sets of type of have size at most and that equality holds if and only if they are caps.

#### 2. Proof of Theorem 1

In this section, first, we briefly recall the basic equations for a -set of with three intersection numbers, and then we will assume that and we will give the proof of Theorem 1.

##### 2.1. The Basic Equations for -Sets with Intersection Numbers 1, , and

Let , , denote the number of planes intersecting in exactly -points (such numbers are called *characters of *).

Double counting gives

From (1) it follows that

##### 2.2. -Sets of with Three Intersection Numbers

From now on, is a -set of with and with intersection numbers 1, , and with respect to the planes.

Since , there are at least two distinct points in . Let and be two points of , and let be the line connecting them. Put . Namely, .

Let denote the number of planes on intersecting in points. Counting via the planes passing through , we have

Being and it follows that ; that is, .

Lemma 3. *If , then , , and is a -cap.*

*Proof. *Let denote the number of points of a line meeting in at least two points. For (5) gives . Thus, and . If , then , and so there would be no plane intersecting in exactly points, against the assumptions. Thus , , , and is a cap.

Lemma 4 (see S. Ferri and O. Ferri [18]). *If , then .*

*Proof. *From it follows that (2) becomes and (3) . Assume that . Let , with a prime number, and an integer. We have that is a divisor of and , but this is impossible since the difference of these numbers is . Thus, .

The proof of Theorem 1 follows from Lemmas 3 and 4.

Let us end with the following easy consequence of Theorem 2.

Corollary 5. *A subset of points of of type has size , and if and only if is a cap of .*

#### Acknowledgment

This research was partially supported by G.N.S.A.G.A. of INdAM.

#### References

- J. Doyen and X. Hubaut:, “Finite regular locally projective spaces,”
*Mathematische Zeitschrift*, vol. 119, pp. 83–88, 1971. - M. J. de Resmini, “On $k$-sets of type $(m,n)$ in a steiner $S(2,\ell ,v)$,”
*Finite Geometries and Designs*, London Mathematical Society Lecture Note Series (No. 49), Cambridge University Press, Cambridge, Mass, USA, 1981. View at Publisher · View at Google Scholar - M. J. de Resmini, “On sets of type $(m,n)$ in BIBDs with $\lambda \ge 2$,” in
*Combinatorial and Geometric Structures and Their Applications (Trento, 1980)*, vol. 14 of*Ann. Discrete Math.*, pp. 183–206, North-Holland, Amsterdam, The Netherlands, 1982. View at MathSciNet - N. Durante, V. Napolitano, and D. Olanda, “On $k$-sets of class $[1,h]$ in a planar space,”
*Atti del Seminario Matematico e Fisico dell'Università di Modena*, vol. 50, pp. 305–312, 2002. - S.-M. Kim, “Sets of type-$(1,n)$ in biplanes,”
*European Journal of Combinatorics*, vol. 25, no. 5, pp. 745–756, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - V. Napolitano and D. Olanda, “Sets of type ${(3,h)}_{2}$ in PG$(3,q)$,”
*Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni*, vol. 22, pp. 1–9, 2012. - V. Napolitano, “On $({q}^{2}+q+1)$-sets of class ${[1,m,n]}_{2}$ in PG$(3,q)$,”
*Electronic Notes in Discrete Math*, vol. 40, pp. 283–287, 2013. - M. Tallini Scafati and G. Tallini,
*Geometria di Galois e teoria dei codici*, C.I.S.U., 1995. - T. Penttila and G. F. Royle, “Sets of type $(m,n)$ in the affine and projective planes of order nine,”
*Designs, Codes and Cryptography*, vol. 6, no. 3, pp. 229–245, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Tallini, “$k$-insiemi e blocking sets in $\text{P}G(\text{2},q)$ ed $\text{A}G(\text{2},q)$,” Quaderni di Geometrie Comb. n. 1 Dipartimento Mat. Univ. l'Aquila, Dicembre 1982.
- G. Tallini, “Giuseppe On line $k$-sets of type $(0,n)$ with respect to pencils of lines in PG$(d,q)$,” in
*Combinatorial and Geometric Structures and Their Applications (Trento, 1980)*, vol. 14 of*Ann. Discrete Math.*, pp. 283–292, North-Holland, Amsterdam, The Netherlands, 1982. View at MathSciNet - G. Tallini, “Teoria dei $k$-insiemi in uno spazio di Galois, Teoria dei codici correttori,”
*Sem. Geom. Combin. Dip. Mat. Univ. Roma La Sapienza, Quaderno*, vol. 65, 1985. - G. Tallini, “On sets of given type in a steiner system,” in
*Finite Geometries (Winnipeg, Man., 1984)*, vol. 103 of*Lecture Notes in Pure and Appl. Math.*, pp. 307–319, Dekker, New York, NY, USA, 1985. View at Zentralblatt MATH · View at MathSciNet - G. Tallini, “Some new results on sets of type $(m,n)$ in projective planes,”
*Journal of Geometry*, vol. 29, no. 2, pp. 191–199, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Tallini Scafati, “The $k$-sets of type $(m,n)$ of an affine space ${A}_{r,q}$,”
*Rendiconti di Matematica. Serie VII*, vol. 1, no. 1, pp. 63–80, 1981. View at MathSciNet - J. Schillewaert, “A characterization of quadrics by intersection numbers,”
*Designs, Codes and Cryptography*, vol. 47, no. 1–3, pp. 165–175, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. Schillewaert and J. A. Thas, “Characterizations of Hermitian varieties by intersection numbers,”
*Designs, Codes and Cryptography*, vol. 50, no. 1, pp. 41–60, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - O. Ferri and S. Ferri, “Su alcuni ${q}^{2}$ insiemi di AG$(3,q)$ a tre caratteri rispetto ai piani,”
*Ratio Mathematica*, vol. 9, pp. 69–72, 1995.