Anticlusters and intersecting families of subsets (Q2641301)

A family \({\mathcal A}\subseteq 2^{[n]}\), \([n]=\{1,2,...,n\}\), is said to be an intersecting family of a family \({\mathcal B}\subseteq 2^{[n]}\) if for any F, \(F'\in {\mathcal A}\) there is \(B\in {\mathcal B}\) such that \(B\subseteq F\cap F'\). In the paper it is shown that if \({\mathcal B}\) consists of all translates of a set \(X\subseteq [n]\) then the maximum size of an intersecting family over \({\mathcal B}\) equals \(2^{n-| X|}\). This proves a special case of a conjecture of \textit{F. Chung, P. Frankl, R. Graham} and \textit{J. Shearer} [ibid. 43, 23-37 (1986; Zbl 0655.05001)].