Fully cotangent subsets of symmetric designs (Q1803362)

Let \(D=(P,B,I)\) be a symmetric \((v,k,\lambda)\)-design of order \(n=k- \lambda\), and \(P_ 1\) a subset of \(P\). The author defines \(P_ 1\) to be a fully cotangent set of \(D\) if there is at least one tangent block to \(P_ 1\) at each point of \(P_ 1\). He proves an upper bound for the size of such a fully cotangent set, namely, \(| P_ 1|\leq ((k- 1)/\lambda)n^{1/2}+1\), with equality if and only if \(P_ 1\) is the set of points of a Hermite subdesign of \(D\), i.e. \(n=s^ 2\) and the subdesign is a 2-\(((s+1)(s^ 2-s+\lambda)/\lambda,s+1,\lambda)\)-design. Consequently, the upper bound above is also an upper bound for an irreducible blocking set. Furthermore, the same bound turns out to be also a bound for the set of absolute points of a planar polarity of a symmetric design. An example is also given of a planar polarity whose set of absolute points does not meet the upper bound.