Sách On randomly generated non-trivially intersecting hypergraphs

1 Introduction
In 1961, Erd˝os, Ko and Rado [5] proved that if 2r 6 n, then the edge set E of an
intersecting r-uniform hypergraph with vertex set V and | V | = n cannot have larger size
than
, moreover if 2r < n, then the only hypergraphs with that many edges are of
the form { e ∈
: v ∈ e} for some fixed v ∈ V . In the past almost five decades, the
area of intersection theorems has been widely studied, but randomized versions of the
Erd˝os-Ko-Rado theorem have only attracted the attention of researchers recently. There
are mainly two approaches to the randomized problem. Balogh, Bohman and Mubayi [2]
considered the problem of finding the largest intersecting hypergraph in the probability
space G
(n, p) of all labeled r-uniform hypergraphs on n vertices where every hyperedge
appears randomly and independently with probability p = p(n). In this paper, we follow
the approach of Bohman et al. [3], [4]. They considered the following process to generate
an intersecting hypergraph by selecting edges sequentially and randomly.