Finite projective spaces and intersecting hypergraphs

Abstract

Let ℱ be a family ofk-subsets on ann-setX andc be a real number 0 <c<1. Suppose that anyt members of ℱ have a common element (t ≧ 2) and every element ofX is contained in at mostc|ℱ| members of ℱ. One of the results in this paper is (Theorem 2.9): If $$c = {{(q^{t - 1} + ... + q + 1)} \mathord{\left/ {\vphantom {{(q^{t - 1} + ... + q + 1)} {(q^t + ... + q + 1)}}} \right. \kern-\nulldelimiterspace} {(q^t + ... + q + 1)}}$$ . whereq is a prime power andn is sufficiently large, (n >n (k, c)) then Open image in new window The corresponding lower bound is given by the following construction. LetY be a (qt + ... +q + 1)-subset ofX andH1,H2, ...,H|Y| the hyperplanes of thet-dimensional projective space of orderq onY. Let ℱ consist of thosek-subsets which intersectY in a hyperplane, i.e., ℱ={F∈(kX): there exists ani, 1≦i≦|Y|, such thatY∩F=Hi}.