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 \([m]\cap [n]\).

令\（m，n，\ ell \）和k为正整数，其中\（\ ell \ ne k \），\（n> 2k \），\（m <2 \ ell \）和\（n = \ max \ {m，n \} \ ge \ ell + k \）。如果\（\ mathcal {F} \）是\（\ left（{\ begin {array} {c} [m] \\ \ \ ell \ end {array}} \ right }）中的一个相交的族\ cup \ left（ {\ begin {array} {c} [n] \\ k \ end {array}} \ right）\），然后

除非\（n = \ ell + k \ ge m \），否则当且仅当\（\ left（{\ begin {array} {c} m-1 \\ \ ell \ end {array}} \ right ）\ ge \ left（{\ begin {array} {c} n-1 \\ k-1 \ end {array}} \ right）\）和\（\ mathcal {F} = \ left（{\ begin { array} {c} [m] \\ \ ell \ end {array}} \ right）\）或\（\ left（{\ begin {array} {c} m-1 \\ \ ell \ end {array} } \ right）\ le \ left（{\ begin {array} {c} n-1 \\ k-1 \ end {array}} \ right）\）和\（\ mathcal {F} \）都由的成员\（\左（{\开始{阵列} {C} [米] \\ \ ELL \ {端阵列}} \右）\杯\左（{\开始{阵列} {C} [N] \ \ k \ end {array}} \ right）\）包含\（[m] \ cap [n] \）的固定元素。