Quasi-symmetric \(\{\) \(\delta\) ,\(\mu\) \(\}\)-biplanes (Q1124113)

Let \(\delta\) and \(\mu\) be non-negative integers, \(\delta\leq \mu\). An incidence structure with point set S and block set \(\beta\) is called a quasi-symmetric \(\{\) \(\delta\),\(\mu\) \(\}\)-biplane if it satisfies the following axioms: (i) Any two distinct points are on exactly two blocks. (ii) Two distinct blocks meet in one or two points and will accordingly be referred to as 1- or 2-secant blocks. (iii) For any pair (p,B), \(p\not\in B\), denote by \(\mu\) (p,B) the number of blocks on p which are 1-secant to B. Then \(\pi\) (p,B)\(\in \{\delta,\mu \}.\) (iv) (Non-degeneracy axiom) There are at least two blocks (points) on any point (block). Given two points (blocks) there is a block (point) not on them. In case \(\delta =0\) such a structure is a semi-symmetric design in the sense of \textit{A. Beutelspacher} [Ann. Discrete Math. 14, 83-116 (1982; Zbl 0499.05019)]. The authors therefore require \(\delta >0\). The main result is the following theorem: Theorem. Suppose (S,\(\beta)\) is a quasi-symmetric \(\{\) \(\delta\),\(\mu\) \(\}\)-biplane with \(0<\delta \leq \mu\). Then either (i) \(\delta =\mu =4\) and (S,\(\beta)\) is the residual design of a biplane, or (ii) \(\delta =2\), \(\mu =4\) and (S,\(\beta)\) is obtained by deleting a block but for one of its points from a biplane. The proof depends on counting arguments.