Intersecting families in $\left( {\begin{array}{c}{[m]}\\ \ell \end{array}}\right) \cup \left( {\begin{array}{c}{[n]}\\ k\end{array}}\right)$

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

$$\begin{aligned} |\mathcal {F}|\le \max \left\{ \left( {\begin{array}{c}m\\ \ell \end{array}}\right) , \left( {\begin{array}{c}m-1\\ \ell -1\end{array}}\right) +\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) \right\} . \end{aligned}$$

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 $[m]\cap [n]$.