The study of intersection problems on families of sets is one of the most important topics in extremal combinatorics. As we all know, the extremal problems involving certain intersection constraints are equivalent to that with the union properties by taking complement of sets. A family of sets is called s-union if the union of any two sets in this family has size at most s. Katona [Acta Math. Hungar. 15 (1964)] provided the maximum size of an s-union family of sets of [n], and he also determined the extremal families up to isomorphism. Recently, Frankl [J. Combin. Theory Ser. B 122 (2017)] sharpened this result by establishing the maximum size of an s-union family that is not a subfamily of the so-called Katona family. In this paper, we determine the maximum size of an s-union family that is neither contained in the Katona family nor in the Frankl family. Moreover, we characterize all extremal families achieving the upper bounds.
@article{10_37236_12706,
author = {Yongtao Li and Biao Wu},
title = {Stabilities for non-uniform \(t\)-intersecting families},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12706},
zbl = {1551.05397},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12706/}
}
TY - JOUR
AU - Yongtao Li
AU - Biao Wu
TI - Stabilities for non-uniform \(t\)-intersecting families
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12706/
DO - 10.37236/12706
ID - 10_37236_12706
ER -
%0 Journal Article
%A Yongtao Li
%A Biao Wu
%T Stabilities for non-uniform \(t\)-intersecting families
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12706/
%R 10.37236/12706
%F 10_37236_12706
Yongtao Li; Biao Wu. Stabilities for non-uniform \(t\)-intersecting families. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12706
