A quest for certain linear spaces on thirty points
A linear space is an incidence structure in which: (i) every line contains at least two points; (ii) there are at least two lines; (iii) every pair of points is joined by a unique line. A line is said to be a k-line if it consists of k points. We write [k] to denote the number of k-lines in a linear space. A linear space \(\Sigma\) is said to be a PBD(X,v) if \(\Sigma\) has v points and if the cardinality of every line of \(\Sigma\) is a member of the set X. In 1983 the authors determinated the existence or non existence of a PBD(L,v) for all v except \(v=30\), where L denotes the set of positive integers that are not factors of 6 [see J. Comb. Theory, Ser. A 34, 266-300 (1983; Zbl 0518.05011)]. In the present paper, the following result is proved: Let \(\Sigma\) be a PBD(L,30). Then \(\Sigma\) has no k-lines \(\ell (k)\) with \(k>8\) and if \(k\geq k'>6\) then \(\ell (k)\cap \ell (k')\neq \emptyset.\) Finally, the line size distribution ([8],[7],[5],[4]) is one of the following 4-tuples: (1,1,14,41), (0,3,24,22), (0,3,15,37), (0,1,27,24), (0,1,24,29), (0,1,15,44).
scientific article; zbMATH DE number 3877207