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

The corresponding lower bound is given by the following construction. LetY be a (q t + ... +q + 1)-subset ofX andH 1,H 2, ...,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∈( X k ): there exists ani, 1≦i≦|Y|, such thatYF=H i }.

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Frankl, P., Füredi, Z. Finite projective spaces and intersecting hypergraphs. Combinatorica 6, 335–354 (1986). https://doi.org/10.1007/BF02579260

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AMS subject classification (1980)

  • 05 B 25
  • 05 C 35