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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
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 (∪s i=1Ai) ∩ ! ∪t i=1Ai+s " ∕= ∅. The celebrated Erd˝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 Erd˝os-Ko-Rado theorem such as the Han and Kohayakawa theorem [Proc. Amer. Math. Soc. 145 (2017), pp. 73–87] which finds the structure of the third largest intersecting family, the Kostochka and Mubayi theorem [Proc. Amer. Math. Soc. 145 (2017), pp. 2311– 2321], and the more recent Kupavskii’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. 269–282]. As another consequence, our result provides some stability results related to the famous Erd˝os matching conjecture.

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