Subsets of PG(n,2) and maximal partial spreads in PG(4,2)

Put theta (n) = #{points in PG(n, 2)} and phi (n) = #{lines in PG(n, 2)}. L et psi be any point-subset of PG(n, 2). It is shown that the sum of L = #{i nternal lines of psi} and L' = #(external lines of psi} is the same for all psi having the same cardinality: THEOREM A If k is defined by k = \psi\ - theta (n-1), then L + L' = phi (n-1) + k(k - 1)/2. (The generalization of this to subsets of PG(n, 3) is also obtained.) Let S be a partial spread of lines in PG(4, 2) and let N denote the number of reguli contained in S. Use of Theorem A gives rise to a simple proof of: THEOREM B If S is maximal then one of the following holds: (i) \S\ = 5, N = 10; (ii) \S\ = 7, N = 4; (iii) \S\ = 9, N = 4. If (i) holds then S is spread in a hyperplane. It is shown that possibility (ii) is realized by precisely three projectively distinct types of partial spread. Explicit examples are also given of four projectively distinct typ es of partial spreads which realize possibility (iii). For one of these typ es, type X, the four reguli have a common line. It is shown that those part ial spreads in PG(4, 2) of size 9 which arise, by a simple construction, fr om a spread in PG(5, 2), are all of type X.