2 research outputs found
Improved bounds on the maximum diversity of intersecting families
A family is called an intersecting family
if for all . If then is called a star. The diversity
of an intersecting family is defined as the minimum number of
-sets in , whose deletion results in a star. In the present
paper, we prove that for any intersecting family has diversity at most , which improves the
previous best bound due to the first author. This result is derived
from some strong bounds concerning the maximum degree of large intersecting
families. Some related results are established as well.Comment: arXiv admin note: substantial text overlap with arXiv:2207.0548
Invitation to intersection problems for finite sets
Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound