My interest here is mainly on shadows of set systems.
For a family $\cal F$ of sets of size $t$, is shadow $\partial \cal F$
consists of all sets of size $t-1$ that are contained in some
set of $\cal F$.
The Kruskal-Katona theorem says that every family of $m$ sets has
a shadow of size at least $k(m)$, where $k(m)$ is explicitly described
but cumbersome to compute.
I used this theorem in
with Kostochka and Talysheva on poset dimension in the subset lattice.
Lovasz replaced $k(m)$ by a much simpler function $l(m)$
to compute.
The Clements-Linstrom and Macauley theorems are analogs of the
Kruskal-Katona result for multisets, and I was able to prove
a Lovasz-type version and include it in a
with Bekmetjev, Brightwell, and Czygrinow on thresholds in the
multiset lattice (analogous to the Bollobas-Thomason theorem for
the subset lattice, which used Lovasz's estimate).
This paper (which proved the existence of pebbling thresholds for every
graph sequence), is included in the special volume
distinguished by the editors as among
the 12 best of the 210 papers published that year.
Currently I am working on shadow and threshold results in other lattices,
as well as Erdős-Ko-Rado type
involving graphs. My most recent works produced
of the famous Erdős-Ko-Rado and Hilton-Milner theorems and
of EKR theory to lattices.
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