## “School of Mathematics”

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Paper   IPM / M / 17082
 School of Mathematics Title: Size and structure of large $((s,t)$-union intersecting families Author(s): Ali Taherkhani Status: Published Journal: Electron. J. Combin. Vol.: 29 Year: 2022 Pages: 1-22 Supported by: IPM
Abstract:
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 (âªs i=1Ai) â© ! âªt i=1Ai+s " â= â. The celebrated ErdËos-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 ErdËos-Ko-Rado theorem such as the Han and Kohayakawa theorem [Proc. Amer. Math. Soc. 145 (2017), pp. 73â87] 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 Kupavskiiâs 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. 269â282]. As another consequence, our result provides some stability results related to the famous ErdËos matching conjecture.