Linear Spaces With Line Range {n-l, n, n+1} and at most n² points

We characterize all finite linear spaces with p ≤ n2points where n ≥ 8 for p ≠ n2 — 1 and n ≥ 23 for p = n2— 1. and the line range is {n - 1, n, n + 1}. All such linear spaces are shown to be embeddable in finite projective planes of order a function of n. We also describe the exceptional linear spaces arising from p < n2 - 1 and n ≥ 4. © 1980, Australian Mathematical Society. All rights reserved.