tailieunhanh - Báo cáo khoa học:Bounds on the Tur´n density of PG(3, 2) a

For n 2, let PG(n, 2) be the finite projective geometry of dimension n over F2, the field of order 2. The elements or points of PG(n, 2) are the one-dimensional vector subspaces of Fn+1 2 ; the lines of PG(n, 2) are the two-dimensional vector subspaces of Fn+1 2 . Each such one-dimensional subspace {0, x} is represented by the non-zero vector x contained in it. For ease of notation, if {e0, e1, . . . , en} is a basis of Fn+1 2 and x is an element of PG(n, 2), then we denote x by a1 . . .as, where x = ea1 +·. | Bounds on the Turan density of PG 3 2 Sebastian M. Cioaba Department of Mathematics Queen s University Kingston Canada sebi@ Submitted Oct 27 2003 Accepted Feb 18 2004 Published Mar 5 2004 MR Subject Classifications 05C35 05D05 Abstract We prove that the Turan density of PG 3 2 is at least 37 and at most 27 . . 1 Introduction For n 2 let PG n 2 be the finite projective geometry of dimension n over F2 the field of order 2. The elements or points of PG n 2 are the one-dimensional vector subspaces of Fn 1 the lines of PG n 2 are the two-dimensional vector subspaces of Fn 1. Each such one-dimensional subspace 0 x is represented by the non-zero vector x contained in it. For ease of notation if e0 e1 . en is a basis of Fn 1 and x is an element of PG n 2 then we denote x by a1. .as where x eai eas is the unique expansion of x in the given basis. For example the element x e0 e2 e3 is denoted 023. For an r-uniform hypergraph F the Turan number ex n F is the maximum number of edges in an r-uniform hypergraph with n vertices not containing a copy of F. The Turan density of an r_l 1 1 1 tnrm II A 1TA lit .i Illi ì ic Trí í 1 - 1 1 111 eX n -X Ạ Q_11 1 1 I ll1 111 1 A1TA I l lil li Till ỈỮ -ill ll 1 1 I I 1I uniiorm hypergraph F is n linin_ 0 Jn . JA 3 uniform hypergraph is also called a triple system. The points and the lines of PG n 2 form a triple system Hn with vertex set V Hn Fn 1 0 and edge set E Hn xyz x y z E V Hn x y z 0 . The Turan number density of PG n 2 is the Turan number density of Hn. It was proved in 1 that the Turan density of PG 2 2 also known as the Fano plane is 4. The exact Turan number of the Fano plane was later determined for n sufficiently large it is ex n PG 2 2 n - L3J f3 This result was proved simultaneously and independently in 2 and 4 . In the following sections we present bounds on the Turan density of PG 3 2 . THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N3 1 2 A lower bound Let G be the triple system on n 1