Abstract
Summary Consider a finite n-dimensional projective space PG(n, s) over a Galois field of order s = ph (where p, h are positive integers and p is a prime characteristic of the field). A set of k distinct points in PG(n, s), no four coplanar, is said to be complete if there exists no other set with kx points with k1 > k. The number of points in a maximal complete set is denoted by m4(n + 1, s). The exact value of m4(n + 1, 2) is known for n ≤ 7. When n ≥ 8, the best upper bound on m4(n + 1, 2) is due to Seiden (1964). It is the purpose of this paper to show that m4(4, s) = s + 1 for s > 4 and to obtain bounds for m4(n + 1, s), n > 3.
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