A generalization of the Katona theorem for cross t-intersecting families
Let X be an n-element set and let k,t\(\leq n\) be integers. Write \(K^ n_ k\) for the number of at least k-size subsets of X. Let \({\mathcal A}\) and \({\mathcal B}\) be families of subsets of X such that \(| A\cap B| \geq t\) for every A in \({\mathcal A}\) and B in \({\mathcal B}\). In this paper it is shown that (i) If \(n+t=2k\) then \(| {\mathcal A}| | {\mathcal B}| \leq (K^ n_ k)^ 2,\) (ii) If \(n+t=2k-1\) then \[ | {\mathcal A}| | {\mathcal B}| \leq \max \{(K^ n_ k+\left( \begin{matrix} n-1\\ k- 1\end{matrix} \right))^ 2,\quad K^ n_ k K^ n_{k-1}\}. \] These results generalize similar results for the case when \({\mathcal A}={\mathcal B}\) of \textit{G. O. H. Katona} [Acta Math. Acad. Sci. Hung. 15, 329-337 (1964; Zbl 0134.251)].
scientific article; zbMATH DE number 4099300