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Paper IPM / M / 17082  


Abstract:  
A family F of ksets on an nset 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 ErdËosKoRado theorem determines the size and structure of the largest
intersecting (or (1, 1)union intersecting) family. Also, the HiltonMilner theorem
determines the size and structure of the second largest (1, 1)union intersecting
family of ksets. 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 ErdËosKoRado
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 ksets 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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