On some Theorems in Combinatorics Relating to Incomplete Block Designs

In this paper we have studied certain combinatorial properties of incomplete block designs and efficient necessary conditions for the existence of affine resolvable balanced incomplete block (b.i.b., for abbreviation) designs. Two theorems give combinatorial properties of certain b.i.b. designs. The well known inequality of Fisher between the number of varieties and number of blocks is shown in this paper to hold under very general conditions. An intrinsic characteristic property of symmetrical b.i.b. designs is given in another theorem. In the last two theorems we have deduced efficient necessary conditions for the existence of affine resolvable b.i.b. designs. Besides these there are some minor results. Utilizing the simple yet very fruitful idea of associating an incidence matrix with a design, all the results are deduced with the help of arguments of algebra of matrices and linear equations. The last theorem requires the use of the celebrated four square theorem of Lagrange and a result due to Legendre in number theory.