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A family F of k-sets on an n-set X is said to be an (s,t)-union intersecting family
if for any A1, . . . , As+t in this family, we have (âªs
i=1Ai) â© !
âªt
i=1Ai+s
" â= â
. The
celebrated ErdËos-Ko-Rado theorem determines the size and structure of the largest
intersecting (or (1, 1)-union intersecting) family. Also, the Hilton-Milner theorem
determines the size and structure of the second largest (1, 1)-union intersecting
family of k-sets. In this paper, for t ! s ! 1 and sufficiently large n, we 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 ErdËos-Ko-Rado
theorem such as the Han and Kohayakawa theorem [Proc. Amer. Math. Soc. 145
(2017), pp. 73â87] which finds the structure of the third largest intersecting family,
the Kostochka and Mubayi theorem [Proc. Amer. Math. Soc. 145 (2017), pp. 2311â
2321], and the more recent Kupavskiiâs theorem [arXiv:1810.009202018 (2018)]
whose both results determine the size and structure of the ith largest intersecting
family of k-sets for i " k + 1. In particular, when s = 1, we confirm a conjecture
of Alishahi and Taherkhani [J. Combin. Theory Ser. A 159 (2018), pp. 269â282].
As another consequence, our result provides some stability results related to the
famous ErdËos matching conjecture.
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