Sperner extensions of affine spaces (Q578578)

Let \(S=(P(S),L(S))\) be an indidence structure, where P(S) denotes the set of points of S and L(S) is a collection of subsets of P(S) called the lines of S. The incidence is the natural inclusion. In the present paper the Sperner spaces \(S=(P(S),L(S))\) are investigated in which the affine space \(AG(3,q)=A=(P(A),L(A))\) can be embedded in the following sense: 1) P(A) is a subset of P(S). 2) Any line of A is contained in a line of S; moreover for any line e of S if \(e'=e\cap P(A)\) is not empty then it is a line of A, called the affine part of e. 3) Let e and f be two lines of S which have non empty affine parts e' and f'. Then they are parallel in S if and only if e' and f' are parallel in A.