Combinatorial Problems

Abstract

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].