Size and Structure of Large $(s,t)$-Union Intersecting Families

A family $F$ of $k$-sets on an $n$-set $X$ is said to be $(s,t)$-union intersecting if for any $A_1,\ldots,A_{s+t}$ in this family, we have $\left(\cup_{i=1}^s A_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ The celebrated Erdős-Ko-Rado theorem determines the size and structure largest (or $(1,1)$-union intersecting) family. Also, Hilton-Milner second $k$-sets. In paper, $t\geq s\geq 1$ sufficiently large $n$, find out some maximal families. Our results are nontrivial extensions recent generalizations such as Han Kohayakawa theorem~[Proc. Amer. Math. Soc. 145 (2017), pp. 73--87] which finds third Kostochka Mubayi 2311--2321], more Kupavskii's [arXiv:1810.009202018, (2018)] whose both determine $i$th $i\leq k+1$. particular, when $s=1$, confirm a conjecture Alishahi Taherkhani [J. Combin. Theory Ser. 159 (2018), 269--282]. As another consequence, our result provides stability related famous Erdős matching conjecture.