10 research outputs found
Supersaturation and stability for forbidden subposet problems
We address a supersaturation problem in the context of forbidden subposets. A
family of sets is said to contain the poset if there is an
injection such that implies . The poset on four elements with is
called butterfly. The maximum size of a family
that does not contain a butterfly is as proved by De Bonis, Katona, and
Swanepoel. We prove that if contains
sets, then it has to contain at least copies of the butterfly provided for some positive . We show by a
construction that this is asymptotically tight and for small values of we
show that the minimum number of butterflies contained in is
exactly
Structure and Supersaturation for Intersecting Families
The extremal problems regarding the maximum possible size of intersecting
families of various combinatorial objects have been extensively studied. In
this paper, we investigate supersaturation extensions, which in this context
ask for the minimum number of disjoint pairs that must appear in families
larger than the extremal threshold. We study the minimum number of disjoint
pairs in families of permutations and in -uniform set families, and
determine the structure of the optimal families. Our main tool is a removal
lemma for disjoint pairs. We also determine the typical structure of
-uniform set families without matchings of size when , and show that almost all -uniform intersecting families on vertex
set are trivial when .Comment: 23 pages + appendi