On the Existence of Almost Uniform Linear Spaces

Abstract

<p>In this thesis we investigate the existence of a particular class f linear space called an almost uniform linear space. An almost uniform linear space is a linear space in which exactly two lines (called long lines) have sizes u and w, respectively, and all other lines (called short lines) have the same size k (k≥2). We determine the necessary conditions for the existence of an almost uniform linear space, in the cases where the long lines intersect (or are disjoint) and have the same size (or distinct sizes).</p> <p>Next, we are interested in establishing the sufficiency of said conditions for almost uniform linear paces in which the short lines all have size two, three, four, or five. If we assume that the short lines all have size two, this follows immediately. Also, we can show that the conditions are sufficient for almost uniform linear spaces in which the short lines have size three and (i) the two long lines intersect and have the same size u, or (ii) the two long lines intersect (or are disjoint) and have sizes u Є {5,7,9} and w Є {7, 9, 13, 15}, where u ≠ w.</p> <p>By generalizing the conditions in (ii), we provide partial answers to the existence question for almost uniform linear spaces in which one long line has size 6t + 5, 6t + 7 or 6t + 0 (t > 0) and the other long line has size w, w > 6t + r (r = 5,7,9). There are only partial solutions for the case of short lines of size four or five.</p>