Some new results on sets of type (m,n) in projective planes
A set K in a projective plane is said to be of type (m,n) if, for each line \(\ell,| K\cap \ell |\) is either m or n. Here \(0\leq m<n\leq q+1\), where q is the order of the plane. Let \(| K| =k\). It follows that \(k^ 2-k[(m+n)(q+1)-q]+mn(q^ 2+q+1)=0\). The author proves that if \(m=1\) and \(q=p^ k(n-1)\) for some prime p, then K is either a Baer subplane or a unital. If \(m=1\) and \(n\geq 4\) then the two solutions for the above equation in k and the value of q can be expressed in terms of \(P_{s-1}(n)\), \(P_ s(n)\), \(P_{s+1}(n)\), where \(P_ s(n)\) satisfies the recurrence relation \[ P_ s(n)=(n- 2)P_{s-1}(n)+ 1- P_{s-2}(n),\text{ where } P_ 0(n)=0\text{ and } P_ 1(n)=1. \] (This last result does not require \(q=p^ k(n-1).)\) Suppose that (1) \((n-m)^ 2\geq q\), (2) g.c.d.\((m,n)=\) g.c.d.\((m-1,n- 1)=1.\) The author shows that then q is a square, \(n=m+\sqrt{q}\) and \(k=m(q+\sqrt{q}+1)\) or \(k=(m+\sqrt{q})(q-\sqrt{q}+1).\)
scientific article; zbMATH DE number 4002924