A completion problem for finite affine planes (Q1105611)

A partial affine plane (PAP) of order n is an \(n^ 2\)-set S of points together with a collection \({\mathcal A}\) of b n-subsets of S called lines such that any two lines meet each other in at most one point. A PAP (S,\({\mathcal A})\) can be completed if there exists an affine plane (S,\({\mathcal A}')\) such that \({\mathcal A}\subseteq {\mathcal A}'\). The valence of a point p is the number of lines containing p. The author finds out conditions for completion of the PAP: 1) \(b>n^ 2\) and the parallelism of lines is an equivalence on \({\mathcal A}\) (Theorem 2.1); 2) \(b=n^ 2+n-e\), \(e<\sqrt{n+1}\) and some point has the valence \(n+1-e\) (Theorem 3.2); 3) \(b=n^ 2+n-e\), \(e<\sqrt{n}\) and some line contains only points of valence \(n+1\) (Theorem 3.3); 4) \(n\geq 2\), \(b=n^ 2+n-1\) (Theorem 4.1); 5) \(n\geq 4\), \(b=n^ 2+n-2\) (Theorem 4.1).