<XML><RECORDS><RECORD><REFERENCE_TYPE>10</REFERENCE_TYPE><REFNUM>8484</REFNUM><AUTHORS><AUTHOR>Miller,A.</AUTHOR></AUTHORS><YEAR>2007</YEAR><TITLE>Triangle-free Linear Spaces on 13 points with line sizes 2,3, and 4</TITLE><PLACE_PUBLISHED>DCS Technical Report Series</PLACE_PUBLISHED><PUBLISHER>Dept of Computing Science, University of Glasgow</PUBLISHER><PAGES>1-12</PAGES><ISBN>TR-2007-235</ISBN><LABEL>Miller:2007:8484</LABEL><KEYWORDS><KEYWORD>linear spaces</KEYWORD></KEYWORDS<ABSTRACT>In this document we consider linear spaces with lines of size 2, 3 and 4 for which there is no triangular set of pairs of points {(a,b), (a,c),(b,c)} in the lines of size 2 and 3 which are not all contained in a single line of size 3. We call these structures triangle-free linear spaces with lines of size 2, 3 and 4 ({2,3,4}-TFLSs). We show that the only {2,3,4}-TFLSs on 13 points are the 13-4-1 design and a unique structure consisting of 10 lines of size 4 and 4 of size 3 (and six lines of size 2). We show how, in the latter case, the lines can be manipulated to produce a 13-4-1 design.</ABSTRACT></RECORD></RECORDS></XML>