A Generalization of Fisherʹs Inequality

In this paper we are concerned with the following conjecture. Conjecture: Let L be a collection ofkpositive integers and A={A1, …, Am} denote a family of subsets of ann-element set such that |Ai∩Aj|∈L, for allAi,Aj∈A, then |A|⩽∑ki=0 ([equation]). In particular, we show this conjecture is true when L consists ofkconsecutive positive integers. This generalizes a well-known inequality of Fisherʹs. Our proof simplifies and extends a recent result of Ramananʹs.