A family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{\left[n\right]}{k}\right)$ is $U(s,q)$ of for any $F_1,\dots ,F_s\in \mathcal{F}$ we have $|F_1\cup \cdots \cup F_s|\le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number smaller than $s$.

In this paper, we find the maximum $\left|\mathcal{F}\right|$ for $\mathcal{F}$ that are $U(s,q)$, provided $n>C(s,q)k$ with moderate $C(s,q)$. In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős–Ko–Rado theorem, which states that for $n>s^{2}k$ the largest family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{\left[n\right]}{k}\right)$ with property $U(s,s(k-1)+1)$ is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in $\left(\genfrac{}{}{0ex}{}{\left[n\right]}{k}\right)$ is $U(s,s(k-1)+1)$.)

We investigate the case $k=3$ more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be 3 extremal families.

一个家族 $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{\left[ñ\right]}{ķ}\right)$ 是 $ü（s，q）$ 的任何 $F_{\mathrm{1个}}，\dots ，F_s\in \mathcal{F}$ 我们有 $|F_{\mathrm{1个}}\cup \cdots \cup F_s|\le q$。这个概念将一个家庭的财产概括为 $Ť$-相交且匹配数小于 $s$。

在本文中，我们找到了最大 $\left|\mathcal{F}\right|$ 为了 $\mathcal{F}$ 那是 $ü（s，q）$， 假如 $ñ>C（s，q）ķ$ 中度 $C（s，q）$。特别是，我们推广了第一位作者关于Erdős匹配猜想的结果，并证明了Erdős-Ko-Rado定理的推广，该定理指出： $ñ>s^{\mathrm{2个}}ķ$ 最大的家庭 $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{\left[ñ\right]}{ķ}\right)$ 有财产 $ü（s，s（ķ-\mathrm{1个}）+\mathrm{1个}）$是星星，尤其是相交。（相反，很容易看出 $\left(\genfrac{}{}{0ex}{}{\left[ñ\right]}{ķ}\right)$ 是 $ü（s，s（ķ-\mathrm{1个}）+\mathrm{1个}）$）

我们对此案进行调查 $ķ=3$ 更彻底地表明，与Erdős匹配猜想不同，总体上可能有3个极端家族。