Block intersection numbers of block designs. II (Q1057273)

The author continues his work on block intersection numbers [Osaka J. Math. 18, 787-799 (1981; Zbl 0478.05012)]. Main results of the present paper are as follows. (1) Let D be a t-(v,k,1) design. D is called block- schematic if, for given blocks \(\alpha\) and \(\beta\), the number of blocks \(\gamma\) such that \(| \gamma \cap \alpha | =a\) and \(| \gamma \cap \beta | =b\) depends only on a, b and \(| \alpha \cap \beta |\). (1) Let \(\epsilon >0\) and \(t\geq 3\). Then there exist finitely many block-schematic D such that \(v<k^{2-\epsilon}\). (ii) Let \(\epsilon >0\) and \(t>(2/\epsilon)+2\). Then there exist finitely many block- schematic D such that \(v>k^{3+\epsilon}\). (2) Let D be a t- (v,k,\(\lambda)\) design. For a block \(\alpha\) of D \(x_ i(\alpha)\) denotes the number of blocks of D intersecting with \(\alpha\) in i points. Let \(c>2\) be real and \(n\geq 1\) and \(l\geq 0\) integers. Then there exist finitely many D such that (i) \(k-t=n\), (ii) \(v\geq ct\), (iii) there exist a block \(\alpha\) and an integer i (0\(\leq i\leq t-1)\) with \(x_ i(\alpha)=l\).