A new Bollobás-type inequality and applications to \(t\)-intersecting families of sets (Q1877683)

The main theorem generalizes a theorem of \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)] and related results of \textit{L. Lovász} [Problem 13.27, pp.\ 80, 456 of Combinatorial problems and exercises (1979; Zbl 0439.05001)], and \textit{Zs. Tuza} [J. Comb. Theory, Ser. B 39, 134--145 (1985; Zbl 0586.05029)]. Theorem 7. {Let \({\mathcal A}=\{(A_i,B_i):i\in I\}\) be a finite collection of distinct pairs of finite sets, and \(t\geq0\) be an integer satisfying (i) for each \(i\in I\), \(| A_i\cap B_i| \leq t\), (ii) if \(i,j\in I\) and \(i\neq j\), then \(| A_i\cap B_j| \geq t\), (iii) if \(i\neq j\) and \(A_i\cap B_i=A_j\cap B_j\), then \(A_i\cap B_j\neq A_i\cap B_i\neq A_j\cap B_i\). Then \[ \sum_{i\in I}\left(\begin{matrix}| A_i\cup B_i| \\ | A_i\backslash B_i| \end{matrix}\right)^{-1}\left(\begin{matrix}| B_i| \\ | A_i\cap B_i| \end{matrix}\right)^{-1}\leq 1\,. \] }