Size and structure of large $(s,t)$-union intersecting families

A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set $X$. An $(s,t)$-union intersecting family is a family of $k$-sets on an $n$-set $X$ such that for any $A_1,\ldots,A_{s+t}$ in this family, $\left(\cup_{i=1}^sA_i\right)\cap\left(\cup_{i=1}^t A_{i+s}\right)\neq \varnothing.$ Let $\ell(\F)$ be the minimum number of sets in $\F$ such that by removing them the resulting subfamily is intersecting. In this paper, for sufficiently large $n$, we characterize the size and structure of $(s,t)$-union intersecting families with maximum possible size and $\ell(\F)\geq s+\beta$. This allows us to find out the size and structure of some large and maximal $(s,t)$-union intersecting families. Our results are nontrivial extensions of some recent generalizations of the Erd{\H o}s-Ko-Rado theorem such as the Han and Kohayakawa theorem 2017 which finds the structure of the third largest intersecting family, the Kostochka and Mubayi theorem 2017, and the more recent Kupavskii's theorem 2018 whose both results determine the size and structure of the $i$th largest intersecting family of $k$-sets for $i\leq k+1$. In particular, we prove that a Hilton-Milner-type stability theorem holds for $(1,t)$-union intersecting families, that indeed, confirms a conjecture of Alishahi and Taherkhani 2018. We extend our results to $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In fact, when $n$ is sufficiently large, we characterize the size and structure of large and maximal $K_{s_1,\ldots,s_{r+1}}$-free subgraphs of Kneser graphs. In particular, when $s_1=\cdots=s_{r+1}=1$ our result provides some stability results related to the famous Erd{\H o}s matching conjecture.