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Erschienen in:
12.09.2020
Intersecting families in \(\left( {\begin{array}{c}{[m]}\\ \ell \end{array}}\right) \cup \left( {\begin{array}{c}{[n]}\\ k\end{array}}\right) \)
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 4/2020
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Abstract
Let \(m,n,\ell \) and k be positive integers with \(\ell \ne k\), \(n>2k\), \(m<2\ell \) and \(n=\max \{m,n\}\ge \ell +k\). If \(\mathcal {F}\) is an intersecting family in \(\left( {\begin{array}{c}[m]\\ \ell \end{array}}\right) \cup \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \), then Unless \(n=\ell +k\ge m\), equality holds if and only if \(\left( {\begin{array}{c}m-1\\ \ell \end{array}}\right) \ge \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \) and \(\mathcal {F}=\left( {\begin{array}{c}[m]\\ \ell \end{array}}\right) \) or \(\left( {\begin{array}{c}m-1\\ \ell \end{array}}\right) \le \left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \) and \(\mathcal {F}\) consists of all members of \(\left( {\begin{array}{c}[m]\\ \ell \end{array}}\right) \cup \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) that contain a fixed element of \([m]\cap [n]\).
$$\begin{aligned} |\mathcal {F}|\le \max \left\{ \left( {\begin{array}{c}m\\ \ell \end{array}}\right) , \left( {\begin{array}{c}m-1\\ \ell -1\end{array}}\right) +\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \right\} . \end{aligned}$$